Rigidity Phenomena of Group Actions on a Class of Nilmanifolds and Anosov R^n Actions Citation Qian, Nantian (1992) Rigidity Phenomena of Group Actions on a Class of Nilmanifolds and Anosov R^n Actions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/M7JX-ZV02. https://resolver.caltech.edu/CaltechTHESIS:07122011-075820059 Abstract An action of a group Г on a manifold M is a homomorphism ρ from Г to Diff(M). ρo is locally rigid if the nearby homomorphism ρ, ρ(γ) = h o ρ0, (γ) 0h^(-1) for some h Є Diff(M) and for all, γ Є Г. In other words, ρ0 is isolated from other actions up to a smooth conjugation. In this thesis we studied some standard group actions on a broader class of manifolds, the free, k-step nilmanifolds N(n, k); we obtained that the standard SL(n, Z) action on N(n,2) is locally rigid for n = 3, and n ≥ 5. We recall that N(n,1) = T^n. Hence, our results are the generalization to the local rigidity result for the standard action on torus T^n. We observed also, for the first time, that for discrete subgroups Aut(n, 2) of a Lie group, which is not even reductive, the action on N(n,2) is deformation-rigid for n = 3, and n ≥ 5. We also investigated the dynamics of Anosov R^n actions and obtained a number of results parallel to those of Anosov diffeomorphisms and flows. E.g., the strong stable (unstable) manifold for a regular element is dense iff the action is weakly mixing (for a volume-preserving action); an Anosov action with no dense, strong stable (unstable) manifold can always be reduced to suspension of the action mentioned above; there are two compatible measures to the Anosov actions. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Katok, Anatole Thesis Committee: Unknown, Unknown Defense Date: 22 April 1992 Record Number: CaltechTHESIS:07122011-075820059 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07122011-075820059 DOI: 10.7907/M7JX-ZV02 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6538 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 12 Jul 2011 15:21 Last Modified: 09 Nov 2022 19:20 Thesis Files Preview PDF - Final Version See Usage Policy. 30MB Repository Staff Only: item control page

Rigidity Phenomena of Group Actions on a Class of Nilmanifolds and Anosov lR n Actions Thesis by Nantian Qian In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1992 (Defended April 22, 1992) Typeset by AMS- TEX 11 Acknowledgements I began my graduate study in a Ph.D. program at the California Institute of Technology in the fall of 1987. It has been a unique and unforgettable time to study in this carefree environment, to be encouraged by professors and fellow students in the Department of Mathematics at Caltech, and to experience the friendship of the people in Pasadena. I am indeed fortunate to have been pursuing my Ph.D. degree under the supervision of Dr. Anatole Katok. His broad knowledge and deep insight into mathematics have greatly influenced me. His constant guidance and encouragement, his wealth of ideas, his patience and his stimulating conversations made this thesis possible. I thank him for all his help in mathematics and beyond. It. is my pleasure to thank Professor Dingjun Lou for guiding my way into mathematics, and also to acknowledge the hospitality of the Department of Mathematics at Pennsylvania State University and the help I received from Professor Gene Wayne during my visit. I am grateful to Yves Benoist, Jim Lewis, Viorel Nitica, Yakov Pesin, Howard Weiss, and Cheng-Bo Yue for generously sharing their ideas, knowledge and time with me. Finally, I express my thanks to my wife Jane for her love and support. 111 Abstract An action of a group r on a manifold M is a homomorphism p from r to Diff(M). po is locally rigid if the nearby homomorphism p, PC,) = h 0 poC,) 0 h- l for some h EDiff(M) and for all, E r. In other words, po is isolated from other actions up to a smooth conjugation. In this thesis we studied some standard group actions on a broader class of manifolds, the free, k-step nilmanifolds N(n, k); we obtained that the standard SL(n, Z) action on N(n,2) is locally rigid for n = 3, and n 2': 5. We recall that N(n,1) = Tn. Hence, our results are the generalization to the local rigidity result for the standard action on torus Tn. We observed also, for the first time, that for discrete subgroups Aut(n, 2) of a Lie group, which is not even reductive, the action on N(n,2) is deformation-rigid for n = 3, and n ? 5. We also investigated the dynamics of Anosov Rn actions and obtained a number of results parallel to those of Anosov diffeomorphisms and flows. E .g., the strong stable (unstable) manifold for a regular element is dense iff the action is weakly mixing (for a volume-preserving action); an Anosov action with no dense, strong stable (unstable) manifold can always be reduced to suspension of the action mentioned above; there are two compatible measures to the Anosov actions. IV Table of Contents I: Rigidity Phenomena of Group Actions on a Class of Nilmanifolds ......................... . ....... 1 §1 Introduction .................... . ................................. 1 §2 A Class of Nilmanifolds and Their Automorphisms ..... . .......... 7 §3 Anosov Elements Generate Diagonal-block SL(n, Z) Actions and Aut( n, k) Actions ................................ 12 §4 Local Rigidity for Diagonal-block SL(n, Z) Actions ..... . ........ 17 §5 Smooth, Local Rigidity of Abelian Group A Actions ............. 18 §6 Deformation Rigidity of Group Actions .................. . .. . .... 35 §7 New Examples of Cartan Actions and Other Anosov Actions on Tori and Limitation of Our Method. . . . . . . . . . . . . . . . . . . . . . . .. 48 II: Anosov 1R n Actions ........................... . ................... 54 §1 Introduction ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54 §2 Suspension Construction of Anosov 1R n Actions .................. 58 §3 Irreducible, Weakly Mixing and Continuous Spectrum .. . ......... 66 §4 Metric Transitivity .............................................. 75 §5 Measures Compatible with 1R n Actions. . . . . . . . . . . . . . . . . . . . . . . . . .. 83 III: References ........................................................ 88 1 PART I: Rigidity Phenomena of Group Actions on a Class of Nilmanifolds 1.Introd uction As observed by R. Zimmer, A. Katok, S. Hurder, R. Spatzier, J. Lewis and other authors, large groups often act on manifolds in a rigid way. Let G be a connected semisimple Lie group with finite center and without compact factors, reG a lattice. r is said to be irreducible in G if r projects densely into G / N for every non-central, normal subgroup N. Also, if G = K AN is the Iwasawa decomposition for G, then we define the real rank of G, lR-rank(G)=dim(A); we say G has higher rank if lR-rank( G);::: 2. The celebrated Margulis superrigidity theorem (see [Ml],[Z2J) reflects the rigidity of r in G when r is a higher-rank lattice in G. If r is any finitely generated discrete group and G is any topological group, we denote R(r, G) the set of all homomorphisms from r to G with the compact/open topology. The topology can also be described as follows (see [Raj). Fix generators 11, ... " k for r and identify R(r, G) with a closed subset of G k via p t-t (P(fl)' ... , p( ,k)); then the topology on R(r, G) is simply the subspace topology inherited from G k • Note that G acts naturally on R(r, G) by conjugation. A homomorphism Po in R(r, G) is said to be locally rigid if its orbit under the action of conjugation in R(r, G) is open. In other words, Po is locally rigid if it is isolated up to a conjugation. So for locally rigid action po and any nearby homomorphism p E R(r, G), there exists agE G such that p(f) = gPo(f)g-1 for every I E r. There are a number of classical local rigidity results for homomorphisms of r to G in case that G is a finite-dimensional Lie group. A. Wei1 [W] observed that 2 if P E R(r,G) such that H1(r,AdG 0 p) = 0, then p is locally rigid. Margulis' result (see [Ml]), that HI(r,p) = 0 for every homomorphism p of r to GL(N, JR.), where r is irreducible in a higher-rank, connected, semisimple algebraic JR.-group without compact factors, gives a practical criterion of the local rigidity. Suppose the group G is Diff(M) for a compact manifold Mj then we call a homomorphism p E R(r,Diff(M)) an action of r on manifold M. R. Zimmer [ZI] initiated a program of understanding the action of r on compact manifolds. The guiding philosophy is that the finite-dimensional, local rigidity phenomena should be reflected in the context of actions on manifolds. Zimmer raised the question of local rigidity for the action of SL(n, Z) on torus 'JI'n, n 2: 3, during the 1984 M.S.R.L workshop on ergodic theory, Lie group, and geometry, and again in his 1986 address to the LC.M. [ZI]. Several recent results have been obtained by Hurder, Katok-Lewis, Hurder-Katok-Lewis-Zimmer as follows. = SL(n,Z) or any subgroup of finite index, n 2: 3. Let Pt E R(r ,DiHt'JI'n)) be a continuous path based at Po = the stanTHEOREM (HURDER[Hu2]) 1.1. Let r dard action by automorphisms. Then there exists a continuous path gt EDiHt'JI'n) such that pte,) = gtpo(f)gt l for all small t and, E r. 0 = SL(n, Z) (Sp(n, Z)) or any subgroup of finite index, n 2: 4 (n 2: 3). Then the standard action of r on 'JI'n THEOREM (KATOK-LEWIS[K-L2]) 1.2. Let r ('JI'2n) is locally rigid. THEOREM 0 (HURDER-KATOK-LEWIS-ZIMMER[H-K-L-Z]) r = SL(n, Z) or any subgroup of finite index, n > 3. tion of r on 'JI'n is locally rigid. 1.3. Let Then the standard ac- 0 In his paper [ZI], R. Zimmer also posted the question of whether every action of a higher-rank lattice of a semisimple Lie group on a compact manifold, which 3 preserves a smooth volume form, comes from algebra in origin; in other words, whether or not the following examples with simple algebraic constructions exhaust all the possibilities: (1) Isometric actions (2) r acts on M = H j A via p, where reG and A C H are lattices, with A co-compact, and p: G ~ H is a homomorphism. (3) r acts on M = HjA, where A is a (necessarily co-compact) lattice in a nilpotent Lie group N, and r is a lattice of G, where G is a semisimple group of automorphisms of N, such that r preserves A. This is the "global rigidity problem." This question has been answered affirmatively in recent papers by Katok-Lewis [K-L1], and by [H-K-L-Z] for some special classes of lattice actions on high-dimensional tori. THEOREM (KATOK-LEWIS [K-L1]) 1.4. Let r be a subgroup of finite index in SL(n,Z), n ~ 4, !vI = Tn, and p E R(r,DintM)) sucb tbat (1) tbere exists a fixed point; i.e., tbere exists Xo E M sucb tbat p({)xo = Xo for every 'Y E r, (2) tbere exists a direct-sum decomposition of Qn as a vector space over Q, and an element >'0 E A = {'Y E rlTVi Vi, z 1, 2} sucb tbat tbe diffeomorpbism p( >'0) is Anosov. Let p* : r ~ G L( n, Z) denote tbe bomomorpbism corresponding to tbe action on HI(M,Z) "'" zn. Tben p is smootbly equivalent to tbe linear action corresponding to P*; i.e., tbere exists a diffeomorpbism h of M, bomotopic to tbe identity, sucb tbat p({) = hp*({)h- l for every 'Y E r. 0 4 THEOREM (HURDER-KATOK-LEWIS-ZIMMER [H-K-L-Z)) 1.5. Cartan actions of higher-rank lattice r on Tn (n ~ 3) are globally rigid if the invariant 1- dimensional foliations for a suitable Abelian subgroup A c r are the intersections of stable foliations for some Anosov elements. 0 In [K-Ll], a remarkable example has been constructed that shows that nonalgebraic lattice actions exist for lattice actions. This example serves as a counterexample to the conjecture that every action of a higher-rank lattice of a semisimpIe Lie group on a compact manifold, which preserving a smooth volume form, should come from algebra in origin. We remark that all the rigidity results mentioned above deal with the actions on high-dimensional tori. Hurder, Katok, Zimmer and other authors conjectured that the rigidity phenomena should also appear for actions on other classes of manifolds. Our work confirm the conjecture for a class of algebraic actions on some nilmanifolds, as we call them N(n, 2). The nilmanifolds considered are a factor space of 2-step, free nilpotent groups associated with n-dimensional vector spaces. The restriction of freeness is not essential. We consider this class of nilmanifolds mainly because we want the groups of automorphisms to be "big" groups in the sense that they contain higher-rank lattices, as well as Anosov elements. We can construct other nilmanifolds of step 2 and the same argument may produce rigidity phenomena there. Our main results are the following two theorems. THEOREM A. Let r = SL( n, Z) or any subgroup of finite index. Then the standard diagonal-block action ofr on N(n, 2), n ~ 3, n i= 4 is locally rigid. 0 Our approach to the problem combines the structure theory for the lattices in higher-rank groups, especially a theorem by Prasad and Raghunathan [Pr-R] 5 about the relation of Cartan subgroups and lattices in semisimple groups; KatokLewis's non-stationary Sternberg linearization [K-L2]; a theory of Holder continuous linearization developed by Hurder-Katok-Lewis-Zimmer [H-K-L-Z]; and the property that there exists an Abelian subgroup in r generated by Anosov elements, which has enough I-dimensional foliations. As a corollary of our approach, we constructed a new class of examples of locally rigid group actions on tori. The construction is based on some simple facts in multilinear algebras. We also mention that we have "topological deformation rigidity" (in the sense of Hurder, see Theorem 1, with gt being a continuous family of homeomorphisms) of the standard action of higher-rank lattice r on N( n, k) for all n 2:: 3, k ::; n - l. Some of them are smoothly rigid. We hope we can prove the "smooth deformation rigidity" and then local rigidity for all cases in the future. In literature, the only deformation-rigidity phenomena are observed for higherrank lattices in semisimple Lie groups. We obtain the first example of the deformation-rigid action of a discrete group in a Lie group, which is not even reductive. THEOREM B. 4. Aut(n, 2) action on N(n,2) is deformation-rigid for n 2:: 3, n I- 0 The proof of this fact is based on a criterion of topological deformation rigidity of an action by Hurder [Hu3], which reduces the question to the calculation of the first cohomology group for the action together with the deformed action. We first show that Aut(n,2) is generated by several copies of SL(n, Z). Then we prove that a cocycle for the whole group action has to be a coboundary because this cocycle, restricted to each copy of SL(n,Z), is a coboundary. We remark that local rigidity for the rigidity of Aut(n,2) action on N(n,2) should also be true, as conjectured by Katok. We hope to solve this problem soon. We would like to mention that not all nilmanifolds support Anosov diffeomor- 6 phisms. For example, Heisenberg nilmanifolds do not have Anosov diffeomorphisms. Yet, it is conjectured by Katok that some natural, higher-rank lattice actions on such nilmanifolds should also be rigid. 7 2. A Class of Nilmanifolds and Their Automorphisms 2.1 Nilmanifolds N(n, k). Let .N be a Lie group and let A be a discrete subgroup such that .N/ A has a finite Haar measure. We call such a discrete subgroup a lattice. In the case that .N/ A is compact, we call the lattice uniform. Let A : .N --+ JV be an automorphism of .N whose restriction to A is an automorphism of A. Then the automorphism A induces a map on N / A. We call the induced map an automorphism of .N/ A. In previous work by Hurder [Hu2],[Hu3) and Katok, Lewis [K-L1],[K-L2], it is clear that the existence of hyperbolicity, especially the existence of plenty of Anosov elements plays an important role in the rigidity phenomena. We recall a conjecture that the only compact manifolds supporting Anosov diffeomorphisms are tori, some nilmanifolds, and some infranilmanifolds [F). Therefore, it is natural to ask whether the rigidity phenomena can be observed for nilmanifolds. A connected, simply connected nilpotent Lie group may be identified with its Lie algebra via the exponential map. The group operation is then given in terms of the operation in the Lie algebra by the Baker-Campbell-Hausdorffformula, and the exponential map is the identity map. We recall the Baker-Campbell-Hausdorff formula as follows (see [V)). Let 9 be a Lie algebra, let X, Y E g. We define 00 C(X : Y) = L cn(X : Y), n=l where Cl(X:Y)=X+Y 1 C2(X : Y) = 2[X, Y) 1 1 C3(X : Y) = 12 [[X, Y], Y) - 12 [[X, Y), X) 8 1 1 C4(X : Y) = - 48 [Y, [X, [X, Y]]] - 48 [X, [Y, [X, Y111 1 (n + l)C n+l(X : Y) = 2[X - Y, Cn(X : Y)]+ + L L K 2p p~1,2p~n [Ck 1 (X : Y), [... [Ck 2P (X : Y),X + Y] .. . ], k 1 , ... ,k 2P >O kl+ .. k 2p=n + where K 2p are some rational numbers. It is clear that when one calculates the cn(X : Y) by the recursion formula, one finds that each cn(X : Y) is a linear combination of the commutators of the form [Zt, [Z2, [... [Zn-l, Zn] ... ]]] with Zi E {X, Y}. Therefore, for nilpotent Lie algebra, the right-hand side of BakerCampbell-Hausdorff formula is a finite sum. For example, for Abelian Lie algebra C(X : Y) = X + Y; for 2-step nilpotent Lie algebra 1 C(X: Y) =X +Y + 2[X,Y]; for 3-step nilpotent Lie algebra 1 1 1 C(X : Y) = X + Y + 2[X, Y] + 12[[X, Yj, Y] - 12 [[X, Yj,Xj; for 4-step nilpotent Lie algebra 1 1 C(X: Y) = X + Y + 2[X, Yj + 12[[X, Y], Yj1 1 - 112 [[X, Y], Xj- 48 [Y, [X, [X, Y]]]- 4 8 [X, [Y, [X, Y]]]. We also remark that after identification of the connected, simply connected nilpotent Lie group with its Lie algebra, the automorphism of the Lie group and the automorphism of the Lie algebra are the same. (see [A-S]) 9 Now we want to construct a nilpotent Lie algebra with the property that the group of its automorphisms is sufficiently large. Let V be an n-dimensional vector space over lR . Let {Xl, X2, ••• x n } be a basis. We define Nk(V) to be the k-step, free nilpotent Lie algebra associated with V (see [A-S]). We let Co := {[[[[Xi, Xj], Xk], ... J, xd} , J 'V' k-Ibrackets and let C:= Z-span of Co. We may also view Nk(V) as a simply connected, connected nilpotent group. We recall an easy result. PROPOSITION (SEE [A-S)) 2.1.1. There exists an integer m E Z such that mC is a uniform lattice for nilpotent group Nk(V). We point out that for k = 1,2,3,4, we may take m = 1,2,12,48, respectively. We will denote the lattice exp(mC) by A, denote the factor space Nk(V)/A by N(n, k), and denote the group of automorphisms of N(n, k) by Aut(n, k). Our main objective is to investigate the group actions on N(n, k). 2.2 Automorphisms of N(n, k). Recall that A : N(n, k) --. N(n, k) is an automorphism if A induces a Lie algebra automorphism A : Nk(V) --. Nk(V) preserving the lattice group A. Namely, A is a linear-space automorphism preserving the bracket operation and A(A) = A. The following proposition gives a full description of what an automorphism of N( n, k) should look like. PROPOSITION 2.2.1. A(O) 0 0 I A(l) 0 A(1) A(2) 2 1 0 A(O) A(l) k-2 A(2) A(O) Aut(n, k) = A(O) k-l 0 0 k-3 0 0 0 A(k-l) - 0 10 where A~O) E SL(n,Z),A~O) E M(ni x n,Z), (ni dimV(i)- dim [[[[V, V], V], ... ]. V]) can be any matrices, and A~') is determined by A}O) for , v I i bra.ckets j < i by the formula A[v, w] = [Av, Aw]. When A~') = 0 for i ~ 1, the automor- phism is said to be in diagonal block. PROOF. Take A E Aut( n, k); A is lower block-triangular because V(i) EI1 V(i+l) EI1 ... EB V(k-l) is invariant under A. A straightforward computation shows that A~') is determined by A}O), j ::; i. D We next establish the fact that all "rational points" are actually periodic points for the automorphisms. LEMMA 2.2.2. Let A E Aut(n, k), x E Nk(n) be a rational point with respect to a basis chosen from Co; i.e., under that basis, x = (12.,12, ... , 1!.lL). Let the least ql q2 qN common multiple of qis be q. Let m be the constant that appeared in Theorem 2.1.1. Then there exists a positive integer t, such that At(x) = X + qm 2 Z, (Zb"" ZN) E ZN. In other words, the point x Z = considered as a point on N(n, k) is a periodic point for A. PROOF. In fact, the orbit of q~2 under An with n ~ 1 is finite (mod ZN). So there exist tl > t2 ~ 1 such that A tl ( q~2 ) = A t2 (q~2 ) mod(ZN). But A as well as A-I preserves ZN, so A t l - t 2( q~2) = q~2 mod(ZN). Therefore, our first assertion follows. To prove that the point x considered as a point on N(n, k) is a periodic point for A, we notice that C(x + qm 2 z, -x) = 2:~=1 cn(x + qm 2 z, -x), and also that C2 ( x m + qm 2 z, -x ) = 21[x + qm 2 z, -x] = 2m[z, -qx] E m(Z-span of Co), so for 2-step nilmanifold the assertion is true. 11 For higher-step nilmanifolds, we can prove this assertion by proving that en ( X + qm 2 z, -x) E m(Z-span of Co), using induction on n. 0 12 3. Anosov Elements Generate Diagonal-block S L( n, Z) Actions and Aut( n, k) Actions In view of Katok-Lewis [K-L1] [K-L2] (see Theorem 1.4), the existence of Anosov elements is important to get a global rigidity result. Also, various known approaches to obtain the rigidity results use heavily the hyperbolicity of certain elements in the groups. For a general nilmanifold, one cannot expect the existence of an Anosov element in the group automorphisms. For example, we can prove that any nilmanifold with dimension ~ 5 does not have Anosov automorphism. Some nilmanifolds of dimension;::: 5, do not have an Anosov element either. We summarize some easy facts below. PROPOSITION 3.1. Tbe following nilmanifolds do not support Anosov Diffeomopbisms: (1) Nilmanifolds witb dimension ~ 5 tbat are not tori; (2) N(n, k) witb n ~ kj (3) Tbe example constructed in [D}, wbicb bas dimension 9. PROOF. (1) is a straightforward computation because the corresponding Lie algebras do not support Anosov diffeomorphisms that preserves a lattice. (2) is true because any automorphisms of the nilmanifolds have eigenvalue 1. (3) the nilpotent Lie group considered in [D] is a nine-dimensional simply connected, connected group having a unipotent group as its automorphism group. It is easy to see that it has a uniform lattice. Hence, there exists no Anosov diffeomorphism for any nilmanifolds obtained from the group as a factor space by a uniform lattice. 0 Nevertheless, we can prove that for our diagonal-block action, there are plenty of Anosov elements for n > k. Actually, the diagonal-block action is a Cartan 13 action when k = 2. First of all, we give a definition. DEFINITION (CARTAN ACTION) 3.2. Let A be a free Abelian group with a given set of generators ~ = {'Yl, ... "n}. (¢>,~) is a Cartan CT-action on the manifold M if (a) 4> : A x M ~ M is a CT-action on A1; (b) each Ii E ~ is 4>-hyperbolic and ¢>bi) has a I-dimensional, strongest stable foliation Faa; (c) the tangential distributions Er = TFsS are pairwise-transversal with their internal direct sum Er EB··· EB E~s ~ Tlv!. Let 4> : r x M ~ M be a CT -action. \Ve say 4> is a Cartan action if there exists a subset of commuting elements ~ = {,I, ... "n} c r, which generate an Abelian subgroup A, such that the restriction of 4> to A is a Cartan CT-action on N!. We call (4)IA,~) a Cartan subaction for 4>. THEOREM 3.3. Diagonal-block SL(n, Z) Action on N(n, 2) is a Cartan action, ifn ~ 3. PROOF. The proof of this theorem is a consequence of Theorem 5.1.5. 0 It is unfortunate, however, that for any k > 2, the diagonal-block SL(n, Z) action on N(n, k) is never a Cartan action. Although the actions are generally not Cartan actions, they have plenty of Anosov automorphisms. The following theorem asserts that the diagonal-block group is actually generated by Anosov automorphisms for k :::; n - 1. THEOREM 3.4. Consider SL(n, Z) as an action on N(n, k). Then SL(n, Z) is generated by Anosov automorphisms, if n ~ J..: + 1. This theorem is a generalization of a result in [K-L2J, which states that SL(n, Z) 14 as an action on Tn is generated by Anosov automorphisms. The proof, though different from that in [K-L2J, is inspired by the argument there. We will first prove the following Lemma. LEMMA 3.5. SL(n, Z) as a matrix group bas generators A}, ... , A p , sucb tbat eacb Ai is byperbolic witb one eigenvalue> 1 and otber eigenvalues baving abso- lute values < 1. PROOF. We first recall that there exists a matrix A = (aij) E SL( n, Z) such that all eigenvalues of A are positive, real numbers with only one eigenvalue Al > l. The remaining eigenvalues are A2, . .. ,An < 1 [K-L2]. It is clear then that A I (1 ~ 1) has one big eigenvalue Ai and n - 1 small eigenvalues. Since Eij are generators of SL(n, Z), and it is clear that if we can prove that EijA' is hyperbolic with one eigenvalue> 1 and that the rest of the eigenvalues have absolute value < 1, we are done. We rewrite EijA' = EijB- 1 Diag{Al, ... , An}B = B-1(BEijB_1)Diag{A}, ... , An}Bj it is clear that it has the same eigenvalues as those of so we have only to prove that Cij has one eigenvalue > 1 and that the rest of eigenvalues have absolute value < l. Without loss of generality, we will prove C}2 has the desired property. 15 bll k22 . . . bl l k2n )) b21 k22 ·· b21 k2n · ... .. . . bn1 k22 ... bn1 k2n .. I I Diag{A 1,···,A n }· We may assume that bl l k21 ~ o. (Otherwise we consider B(I - e12)B- 1 instead; recall that (I + el2)-l = 1- e12.) Let hn !In) . , Inn In2 then 111 ~ 1 and C 12 = BEI2 B- 1Diag{ AI, . .. , An} 111 Ai 121 A{ · ( ·· I nl Ai 122 A~ !I2A~ !InA~) hnA~ . . .. . I n2A~ . .. .. I nnA~ Consider h1Ai !InA~ hn'\~ Inl Ai Inn'\~ - ,\ /llAi - A det( C 12 - AI) = det ( : ) !In(A~/ AD hn(A~/ AD ) Inn(A~/ A{\ - (A/ AD . 16 It is an easy exercise to prove that for sufficiently large 1, one of the eigenvalues satisfies I>'/>'~ - /111 < /11/2 or /11/2 < >./>.~ < 3/11 /2; the rest of the eigenvalues satisfy 1>'/ >'~I < /11/ 2. Furthermore, it is a straightforward calculation that those eigenvalues with 1>'/>'11 < /11/2 actually satisfy 1>'1 < 1. Indeed, /H>.i - >. !t2)..~ ... hl>'i h2>'~ -).. ... !tn>.~) hn>'~ det.. ( /n~>'{ /n~)..~ - /nn>'~ . :: - >. = (-It(>.n - >'i(f11 +Cl(l))>.n-l + >'i(c2(1))>.n-2 + ... + >'i(l/>.D); suppose that 1 < 1>'1 < (f11/2)>'i; then I>.n - >.i (f11 + cl(l))>.n-l + >.i (C2(l))>.n-2 + ... + >.i (1/ >.~ ))1 ~ I>.n - (fll>.D>.n-11-I>.icl(l)>.n-l + >'i(c2(1))>.n-2 + ... + >'i(1/>'i))1 ~ I>' - (fll>.DII>.n-11-I>'ie:1(1) + )..i(c2(1)) + ... + 1/>.D)II>.n-11 > O. We get a contradiction. 0 The theorem is then a corollary of this Lemma. We remark, however, that we do not have the diagonalizability of those Anosov generators. It is conceivable that we should have diagonalizable, hyperbolic generators. But we cannot prove this fact in this paper. Remark 3.6. · It is easy to prove that Aut(n, k) is generated by Anosov elements, if n ~ k + 1. We omit the proof. 17 4. Local Rigidity for Diagonal-block SL(n, Z) Actions To prove the local rigidity for SL(n.Z) (or any subgroup of it with a finite index) diagonal-block action, we want to utilize a powerful machinery by [H-K-LZ], which gives that for every finitely indexed subgroup r c SL(n.71), the Cartan action can be measurably linearized up to a finite covering. Therefore, we could use the result there to conclude the rigidity of SL(n.71) (or any subgroup of it with finite index) diagonal-block action. We summarize the result of [H-K-L-Z] in the following theorem suitable for our purpose. THEOREM (HURDER, KATOK, LEWIS, ZIMMER) 4.1. Let r c SL(n,71) be a subgroup of finite index; let 0: be a Cartan action of r on ]vI. Then there exists a normal subgroup r' c r of finite index, a finite Galois covering M' ~ ]vI, a lift ~, and measurable vector fields {WI, . .. , Wk}, k = di mM on M', such that the cocycle Do: with respect to the framing {WI, .. . , Wk} is given almost everywhere by a representation p : r' ~ SL(k, 7l). 0 Combining this theorem with the theorem in the next section, we can prove the following theorem. THEOREM A. The standard block action of finite indexed subgroup r c SL(n,71) on N(n,2) is locally rigid, if n ~ 3, n =1= 4. PROOF. The action we consider is a Cartan action; a small perturbation will also be Cartan. Use the fact that we have an Abelian subgroup A; the restricted action is smoothly conjugate to the original linear action (see the next section). Thus we may assume that the perturbation of the action keeps A intact. Then the argument of the proof of Lemma 8 in [H-K-L-Z] goes through without any modification. We conclude that the action of r on N(n, 2) is smoothly conjugate to a linear action. 0 18 5. Smooth, Local Rigidity of Abelian Group A Actions In this section, we investigate the smoothness of topological conjugacy. vVe will prove that for n ;::: 3, n "# 4, topological conjugacy of the diagonal-block actions on N(n,2) are, in fact, smooth. This fact is actually proved by showing that for a certain Abelian group, the action is locally smoothly rigid. 5.1 Density of Exponents. Let r C SL(n, Z) be a subgroup of finite index. Take an Abelian subgroup A c r such that A can be diagonalized over JR. The existence of such a group was established in [R-P]. We also may assume that there exists a splitting Cartan subgroup H of SL(n,JR) and A C HO n r such that HO / A is compact. This result was utilized in [K-L2] to prove the smoothness of topological conjugacy. We summarize this result as a theorem below. THEOREM 5.1.1. There exists a splitting Cartan subgroup H C SL(n, JR), a subgroup A C HO n r, such that (1) HO /A is compact; (2) if we denote Ai(h) to be the i-th eigenvalue of h E HO (for a fixed basis such that all h E HO are diagonal), then A(:= Al x A2 X ••• X An): HO ---+ V1 (:= {(Xl,X2, .•• ,x n ) E R+nlxIX2 "'Xn = I}) is an isomorphism. (Consequently, In(.) : HO ---+ In( VI) is an isomorphism. We will sometimes abuse the notation to say these two isomorphisms are the same and denote them by A : HO ---+ JRn-l = {Yl + Y2 + ... + Yn = O}). Next, we want to prove the existence of Anosov elements in any subgroup r c SL(n, Z) of finite index. LEMMA 5.1.2. If A E SL(N,Z) is diagonalizable, i.e. 19 then as an automorphism of N(n, k), A can be diagonalized as PROOF:. Let VI, V2, ... ,Vn be different eigenvectors for A : R n - t R n; then [[[[Vi, Vi], Vk], . .. ], v,] spans Nk(Rn), which are also eigenvectors for the induced automorphism of N(n, k), and our assertion then is clear. 0 Remark 5.1.3. Some A!l .. . A~l may not be eigenvalues. For example, A; is not an eigenvalue. But all eigenvalues of the induced automorphisms have this form. 0 We will have the existence of Anosov elements in A considered as a subgroup of the group of automorphisms in the next theorem. In what follows in this section, we will denote by A the induced automorphism on N( n, k) from a matrix A E SL(n, Z). We will always assume that k :::; n - 1. THEOREM 5.1.4. There exists an element A E A such that if AI, A2, ... ,An are n different eigenvalues of A as a matrix in SL(n, Z), then Ai, AiAj, ... , A~l A;2 ... A~n, i l + i2 + ... + in :::; k :::; n - 1 are positive, different from 1, dif- ferent from one another. PROOF. Using notation in Theorem 5.1.1, we construct an open set in R n - l as follows. Let m be a positive integer; define n {ilYI + i 2Y2 + .. . + inYn =J O}; litl+·· ·+linl~m il,i2, ... ,i n are not all equal it is the complement set of finite hyperplanes. Every connected component, say Do, have an arbitrarily large diameter. So Do contains a fundamental domain for A in Ho (view A as a lattice of Rn-I,H o as Rn-l). So AnD =J 4>. Then any element in the intersection can be taken as A. 0 20 Fix an eigenvector v for A, A as in last Theorem; v is then a common eigenvector for all B, B E A. Denote E(v):= {eigenvelues for all B corresponding to the eigenvector v,B E A}; we hope the following is true: E(v)=lR. Till this writing, we can obtain only this result for N(n, k) for certain n, k. We do not know whether it is true in general. THEOREM 5.1.5. Let n = dim(V), k ::; n - 1 be the step of our nilpotent algebra. H n =I 4,6,8, ... , 2( k + 1), then E( v) = R for all common eigenvectors of A (A E A). To prove this theorem, we need two preliminary results, which have their own interest. We state them as two propositions. PROPOSITION eigenvalues. 5.1.6. Let A : '1['n ~ '1['n be an automorphism with different Let f(>.) be the characteristic polynomial of A. Let p(>.) be an irreducible factor of f( >. ) (over Q). Then (1) there exists a subtorus '1['k such that A('1['k) = '1['k, and the characteristic polynomial of A hrk = pC>'); (2) any automorphism B such that AB = BA, satisfies B('1['k) = '1['k. PROOF. (2) is clear from the fact that all the eigenvalues of A are different. To prove (1), let f( >.) = p( >. )h( >.) be a decomposition of f over Q; then we may take p(>.), h(>.) E Z[>']. (See, for example, T. Hungerford [H]: Algebra, p.163, Lemma 6.13). Since p(>.), h(>.) do 21 not have common zeros, we obtain (p, h) = 1. We choose p', h' E Z[>.] such that 1 = p'p + h'h. (See T. Hungerford [H], p.140, Theorem 3.11 (ii) for a proof of this fact.) Therefore, x = p'(A)p(A)(x) + h'(A)h(A)(x) for all x E '['k. Define Ker(p(A)) = {x : p(A)x = a}, Ker(h(A)) = {x: h(A)x = a}. They are closed A-invariant subgroups, and it is clear that K er(p(A)) + K er(h(A)) = '['n. Let y E K er(p(A)) n K er(h(A)); then y = p'(A)p(A)y + h'(A)h(A)y. Hence, from the fact that p(A)y = a and h(A)y = 0, we have y = O. In other words, K er(p(A)) n K er(h(A)) = O. Therefore, ,[,n = K er(p(A)) EEl K er(h(A)). But then it is easy to see that the closed subgroups Ker(p(A)) and Ker(h(A)) are subtori of ,[,n and that they intersect transversely, and the intersection is only one point. 22 Now consider Alker(p(A» : K er(p(A)) -+ K er(p(A)), an automorphism of K er(p(A)), so it has integer-matrix representation for some basis. Therefore, the characteristic polynomial p(..\) belongs to Z["\]. Since p(AIKer(p(A») = 0, it is clear then that the minimal polynomial of AIKer(p(A» divides p(..\). But p(..\)lf(..\), so there are no multiple roots for P(.A). SO the minimal polynomial of AIKer(p(A» is exactly the characteristic polynomial of AIKer(p(A»; hence, p(..\)lp(..\). But p(..\) is irreducible over Q, so p(..\) = p(..\). o To state the next proposition, we need some easy facts from multilinear algebra. We give a brief summary here. See also [N]. Let f : M -+ N be a linear map between vector spaces M, N, let S(J\lI) , S(N) be the symmetric tensor algebras of M, N. Then f has a unique extension to a homomorphism S(f) = ffiSp(f) of graded algebra S(M) = ffiSp(M) to S(N) = ffiSp(N). Moreover, if mI, ... , mp E M, then If 9 : K -+ M is another linear map between real vector spaces K, M, then S(f) 0 S(g) = S(f 0 g) It is clear then that if M = N = K := V and f : V -+ V is an isomorphism, then S(f) as well as Sp(f) is an isomorphism. In other words; any matrix subgroup of GL(n, lR) acts on S(V) as well as Sp(V) in a "standard way." 23 Let V be an n-dimensional vector space; then we may consider S(V) (Sp(V)) as a real vector space. Fix a basis {el' ... , en} of V; then {ei 1 (1 :::; i l :::; n); ei 1 ei 2 (1 :::; i l :::; i2 :::; n); ... ; ei l . . . ei" (1 :::; i l :::; ... :::; in :::; n)} is a basis for S(V), and {ei l ••• ei p (1 :::; i l :::; ... :::; ip :::; n)} is a basis of Sp(V). Let us take a lattice generated by the integer points with respect to the chosen basis. Then any subgroup r of SL(n, Z) induces an action preserving the lattice. So it factors through a toral action. It is clear that if A is a diagonalizable subgroup, then the SeA) is also diagonalizable; if A is an element in A with n different eigenvalues (A as a matrix in SL(n, /Z)) >'1, >'2, ... ,>'n, then >'i, >'j>'j, ... ,>'~l >,;2 ... >,~n, 1 :::; i 1 + i2 + ... + in :::; k, ij ~ 0 are all eigenvalues for SeA). For some special A. as in Theorem 5.1.4, they are positive, different from 1, different from one another. PROPOSITION 5.1.7. Let {VI," . , v n } be common eigenvectors for A as a ma- trix subgroup of SL(n, /Z). Let v be a common eigenvector for A as a subgroup of induced, nipotent algebra automol'pbisms. Let v be obtained by bracket operations from {Vl, ... ,v n }. Tben E(v) =lR,ifn =J:.4,6,S, ... ,'2k. PROOF. We actually prove more. ""Ve will prove that for any common eigen- vector v of Sp(A), v = viI ... v: q , E(l') = lR. if eyery irreducible factor of the characteristic polynomial of Sp(A) has a degree ~ 3. Take A as in Theorem 5.1.4; let be the eigenvalues of A. Take a Q-irreducible factor p(>.) of the characteristic polynomial of Sp(A). Let the corresponding invariant space be virr, which has the dimension r =dim(p(>.)). Also let the corresponding torus be yr. 24 Let - < 1 < 'rJl+ < 'rJ2+< • • • < 'rJr2 + O < rlt- < 'rJ2- < ... < 'rJ r1 be eigenvalues of Sp(A)IVirr. It is then clear that -+ -+ -+ +_\'jl\lj2 ,'jn - Al A2 ••• An 'rJj L: i; :::; r (because virr has dimension r). Go back to the notation of Theorem 5.1.2; we know that n n D = (nj(l: itYt > O}) n cnj t=l {L iitYt < O}) -I- <P t=l and that D is an open set of lR n-l. An ED::} D is unbounded. Take a connected domain D( A) containing A; take an unbounded side L of DC A) that does not intersect any other sides. It is in fact a part of a hyperplane (codimension 1, affine subspace). Claim: If L = {L:t itYt = O}, then for all s E L - {other sides}, there exists a neighborhood U( s) of s, and s± E UC s), such that s+ belongs to L:t itYt :2: OJ and s- belongs to L:t iitYt :::; O. (This is a obvious.) The consequence of this is that A contains at least two elements A + , A -, as automorphisms of ']I'r, the signs of the logarithm of the ordered eigenvelues of which differ in the following fashion: ++ ... ++-_ ... -++ ... +--- ... _We claim also that if A is taken as in Theorem 5.1.2, then every 1-dimensional eigenspace of Sp(A)IVirr project densely to ']I'r. This is because pC>') is irreducible. 25 Indeed, if it is not the case, then the closure of the projection of the 1-dimensional eigenspace of virr would contain an A-invariant subtorusj the characteristic polynomial of the restriction of Sp(A) to that smaller torus will then divide peA). Now we can prove the density of E(v) in R. Otherwise, E(v) n R+ is a infinite cyclic subgroup of R+. This implies that Sp(A+t(v) = Sp(A-)m(v) for some integers m, n. Because that every 1-dimensional eigenspace of Sp(A)IVirr projects densely to ']['r, we know that the above identity holds for all t E ']['rj i.e., Sp(A+)n(t) = Sp(A _)m(t). this will contradict the pattern in (*). It is clear that when n =j:. 4,6,8, ... , 2k, the characteristic polynomial of Sp(A)lsp(v) cannot have factors with a degree < 3. (This is because the constant term of the polynomial has to be 1, and any product of two eigenvalues cannot be 1.) We thus finish our proof. 0 5.2 Smoothness of topological conjugacy. In this section, we will abuse the notation not to distinguish the elements of S L( n, Z) and the induced automorphisms on N(n,k). Using Katok-Lewis's non-stationary Sternberg linearization theorem and a similar argument in [K-L2], we may prove the following theorem. THEOREM 5.2.1. A action on N(n,2) is locally rigid, if n =j:. 4, n ~ 3. COROLLARY 5.2.2. Topological conjugacies for small perturbations of group actions r on N(n, 2) are smooth as long as r contains a Cartan subgroup A as in Theorem 5.1.2. We are going to follow the idea in [K-L2] to prove the local rigidity of the A action on N(n,2). First of all, we give a proposition that provides simultaneous diagonalization of a small perturbation of the A action. The same result for the action on torus is essentially given by [Pa-V], which is utilized in [K-L2]; their result 26 is a little bit stronger than ours, but for the purpose of proving local rigidity, the result below is sufficient. We will modify their argument to fit our situation. PROPOSITION 5.2.3. Let A be an Abelian subgroup of SL(n, Z) acting on N(n, k) witb n ~ k + 1. Assume tbat tbere exists an Anosov element A in A. Let p( A) be a small perturbation of tbe action sucb tbat tbere exists a conjugacy f (close to identity) between A and p( A). H tbe perturbation is small enougb, tben ! is a conjugacy between all B E A and pCB). PROOF. Since !-1 peA)! = A, we have !-1 p(A)p(B)f = Af- 1 p(B)f for all B E A. Let us fix no generators AI, A 2 , ••• ,Ana for A; fix a Remannian metric d on N(n, k). Assume that the perturbation is so small that where Co is the expansive constant for A. From we obtain AA;l 1-1 p(Ai)f = A;l 1-1 p(Ai)f A. Let Cj := A;l !-1 p(Ai)!; then for all x E N(n, k - 1); hence 27 for all x E N(n, k - 1) and m E Z. This forces So hence, o REMARK. Our argument here has the advantage that we actually can prove more; i.e., we can prove that for any finitely generated, discrete Abelian group action on a compact manifold with one expansive element, a small conjugacy of this element is the conjugacy of the action. Now we will apply Katok-Lewis's Non-stationary Sternberg Linearization Theorem [K-L2] to conclude that the conjugacy is actually smooth along the 1dimensional foliations. THEOREM(KATOK-LEWIS' NON-STATIONARY STERNBERG LINEARIZATION) 5.2.4. Let M be a compact manifold,.c = M x ~, the trivial. real line bundle over M, and let F:.c -+ .c,(x,t) ~ (J(x),Fx(t)), with Fx a Coo diffeomorphism of ~ for each x EM; satisfy (1) Fx(O) = 0 for every x E lvI (F preserves the zero section); (2) 0 < F~ < 1 for every x E JVI, t E ~, and (3) x ~ Fx is a continuous map M -+ COO(~). Then there exists a unique reparameterization G: .c -+.c, (x, y) ~ (x, Gx(t)), 28 such that (1) each G x is a Coo diffeomorphism ofR; (2) Gx(O) = 0, G~(O) = 1 for every x E M; (3) x ~ G x is a continuous map M -+ COO(R), and (4) GFG-l(X,t) = (f(x),F~(O)t) for every x E M, t E R. Fix an A E A, such that all the eigenvalues of A as an automorphism of N(n, k) are different. Fix a I-dimensional foliation F of N(n,2) corresponding to an eigenvector v of A; hence v is a common eigenvector for A. Hence the I-dimensional foliation F of N( n, 2) is an invariant foliation for A. It will be proved in Lemma 5.2.5 that this I-dimensional foliation is the intersection of several stable foliations for some Anosov automorphisms. As an important corollary of this lemma, if we take a small perturbation of the action A on N(n, 2), then the foliation F persists, and also the individual leaf of the perturbed foliation F' is Coo manifold. The perturbed foliation F' is Holder foliation and varies continuously (because the foliation is strong stable foliation for some Anosov element, then the results follow from [B-P],[Sh]). The topological conjugacy f as in Proposition 5.2.3 carries the leaves of F to those of F'. LEMMA 5.2.5. The i-dimensional foliation F of N(n, 2) with n ~ 3 is the intersection of several stable foliations for some Anosov automorphisms in A of N(n,2). The action of A on N(n,2) is a Cartan action. PROOF. It is clear that leaves of F are the images under 7r of lines in the universal covering RN of N(n, 2) parallel to VA, which is the eigenvector corresponding to the eigenvalue A. Therefore, it is sufficient to prove that in the universal cover, or in the Lie algebra considered as a linear vector space over IR, every I-dimensional, common eigenspace space of A is the intersection of several stable vector spaces 29 of Anosov elements in A. Let the Lie algebra 9 be associated with n-dimensional vector space V. Let A have n-different, common I-dimensional eigenspaces. Take an ordered basis {VI, ... , v n }; each Vi generates an eigenspace. It is clear that all I-dimensional, common eigenspaces of A are generated by Vi; [Vj, Vk] for i,j, k = 1, ... , n,j -=1= k. We first show that Vi is the intersection of several stable manifolds of Anosov elements in Aut(n,2). Without loss of generality, we let i = 1. Take A E A such that as a matrix in SL(n, Z), it has one eigenvalue (corresponding to eigenvector VI) Al < 1, and other eigenvalues (corresponding to eigenvectors Vi) Ai > 1. In other words, A has ordered eigenvalues AI, ... ,An, satisfying Therefore, the stable vector space WS(A) for A as an automorphism on 9 is spanned by VI; [V!, V2], ... , [VI, v n }. Next, we take A' E A such that as a matrix in SL(n, Z), A' has ordered eigenvalues A~, A~, ... , A~, satisfying o < A~,A~ < 1 < A~, i =1= 1,2. The existence of such A' is a corollary of Theorem 5.1.1. We may assume also that A~A~ > 1, A~ A~ < 1, i -=1= 1,2. (For example, we may consider AP(A')q to be our new A'.) Hence, the stable vector space WS(A') for A as an automorphism on 9 is spanned by VI, V2; [VI, V2], . .. , [VI, v n ]. Similarly, we may find B E A, such that the stable vector space WS(B) for B as an automorphism on 9 is spanned by VI, V2; [V2, VI]' [V2' 1)3], ... ,[V2, v n ]. So WS(A) n WS(A') n WS(B) is spanned by VI, [VI, V2]. The same argument may imply that the space spanned by VI, [VI, V3] is the intersection of some stable vector spaces for some elements in A. So the space 30 spanned by VI is the intersection of some stable vector spaces for some elements in A. Next, we will show that [Vi,Vi], i =1= j is the intersection of several stable manifolds of Anosov elements in Aut(n,2). Without loss of generality, we let i = 1, j = 2. We already know that the space spanned by VI; [VI, V2], ... , [VI, v n ] is the stable vector space WS(A) for some A E A. Similarly, the space spanned by V2; [V2' VI], [V2' V3], ... , [V2, Vn ] is the stable vector space WS(C) for some C E A. So the space spanned by [VI, V2] is the intersection of several stable vector spaces of Anosov elements in Aut(n, 2). The other statement in the lemma is clear. 0 The leaves of both F and F' inherit natural Riemannian metrics as submanifolds of N(n,2). Let VA be the vector in g that determines F. For each x E N(n,2), let C/>X : R. ~ F(x) denote the arc-length parameterization based at x, oriented so that VA points in the positive direction; i.e., c/>x(O) = x, the distance along F(x) between x and c/>x(t) E F(x) is t, and (VA' (c/>x)*(d/dt)) > 0 (standard inner product on Tx(N(n, 2)) ~ JRN). Define 'Jx : R. - F(x) similarly, oriented so that 'Jj/x) 0 f 0 C/>X : R. ~ R. is an orientation-preserving homeomorphism. Recall that a family {WxJxEM of k-dimensional Coo submanifolds of lvI is said to vary continuously if for each x E M there exists a neighborhood U of x in M and a continuous map c/> : M - coo(Dk, M) such that c/>x maps Dk diffeomorphically onto a neighborhood centered at x in W x , where Dk denotes the unit disk in IRk. It is well-known (see Shub [ShD that the strong, stable foliation varies continuously. By construction, c/>x : R. every x E N(n,2). F( x) and c/>x : R. - Fare diffeomorphisms for Let £ x R. denote the trivial line bundle over N(n,2). It follows easily from the continuous varying of the two foliations that c/> : £ - N(n, 2), (x, t) ~ c/>x(t) and 'J : £ - N(n, 2), (x, t) ~ 'Jx(t) are continuous, and 31 that x 1-+ ¢>x, x 1-+ ¢x are continuous maps N(n,2) ---+ COO(JR, N(n, 2)). Let f = peA) EDiff(N(n,2)). Extend f and h to transformations on £ in the obvious way F: £ ---+ £, (x ,t) 1-+ U(x),Fx(t)) and H: £ ---+ £, (x,t) 1-+ (h(x),Hx(t)), so that ¢(F(x, t)) = f(¢(x, t)) and ¢(H(x , t)) = h( ¢(x , t)). Then F and H are continuous, Fx E COO(JR) for each x E N(n,2), 0 < F~(t) < 1 for every x E N(n,2), t E JR, and x 1-+ Fx is a continuous map N(n,2) ---+ COO(JR). In fact, Fx(t) is the length between f(x) and f(¢x(t)). Next we will show that Hx E COO(JR) and x 1-+ Hx are continuous. The argument is identical to that of [K-L2]. For the sake of completeness, we will repeat the argument below. And also we remark that this is the only place that we need the density of the exponents. By Theorem 5.2.4, there exists a unique continuous linearization G : £ ---+ £, (x, t) 1-+ (x , Gx(t)) such that (1) G x E COO(JR) with G~(O) = 1 for every :z; E N(n, 2); (2) N( n, 2) ---+ COO(JR), ;Z; 1-+ G x is continuous; (3) GFG- 1 (x,t) = (f(x),F~(O)t) for every x E N(n,2) and t E JR. LEMMA 5.2.6. Suppose that p E N(n , 2) is rational (hence a periodic point for the standard action of every element the BSL(n, Z)). Then Gh(p) 0 Hplll+ : IR+ ---+ IR+ has the form Gh(p) 0 Hp(t) = cptVp for some cP ' Vp > O. 32 PROOF. Since A is Abelian, it follows from the uniqueness of G that G simulta- neously linearizes the transformations on £, corresponding to p( A) for each A E A. From the fact that the eigenvalue set E( v) is dense in R, we can find B , C E A such that Av(B) = f3, Av( C) = , with f3" > 1, such that f3" generate a dense subgroup in R+. By replacing B and C with appropriate powers, we may assume that p is a fixed point for the action of both Band C. Let r = p(B), s = p(C), and define corresponding R, S : £, - t £, as before, so that <pR = rJ; and J;s = sJ;. Then where f3x = R~(O), '7x = S~(O). In particular, since h(p) is fixed by rand s, with f3 = f3 h(p), '7 = '7h(p) , Also, since h intertwines p and the standard action, Let Then we have shown that for every t E lR +, t/J(f3t) = f3w(t) and t/J(rt) = '7t/J(t). By construction, t/J is an orientation-preserving homeomorphism. Let c = t/J(l). Then t/J(f3 k,') = cjjk'7' for every k, I E Z . Hence, 33 is an order-preserving, continuous map between these two subsets of ~+. If we denote P = pOil and '1 = , 0i 2, then it is easy to see that 0:1 = 0:2 := v. Hence But this set is dense in JR+ and t/J is continuous; hence t/J(t) = ct" for every () E JR+. o Now for each x E N(n,2), set 'l/Jx = Gh(x) 0 Hx II : 1-+ lR+, 1= [0,1]. Since I is compact and GoHIN(n,2)xI is continuous, it follows that N(n , 2) -+ C°(1), x t-t 'l/Jx is continuous with respect to the uniform topology on C°(1). But 'l/Jp(t) = cpt~ for a dense set of p E N(n,2), and vp = log(!p, which is continuous in p on a dense log (3p subset, so p t-t c p , and p t-t vp must extend to continuous functions N(n,2) -+ lR+ such that 'l/Jx(t) = cxt"'" for every x E N(n,2). An entirely analogous argument works with - I = [-1,0] in place of I and lR- in place of lR+. Also, we can replace I with any compact interval [0, T]. Thus, we have proved the following: LEMMA 5.2.7. There exist continuous functions c±, v± : N(n,2) -+ lR+ such that for every x E N(n,2), Gh(x) 0 Hx : lR -+ lR has the form Gh(x) 0 Hx(t) = { c+t"; x _ -c-; It I"", t~O t ~ 0. Now for each x E N(n, 2), Gh(x) oHx is smooth away from 0, and Gh(x) is a Coo diffeomorphism; hence Hx is smooth away from 0. But </> maps N(n, 2) x (lR - {O}) onto N( n, 2), so this implies that h is Coo along each leaf of F; more precisely, hl.F(x) : F(x) -+ F(h(x» is Coo for every x E N(n,2). Thus, Gh(x) 0 Hx must be smooth at 0. So c,% = c-;, and v"1" = v; = 1 for every x E N(n,2). We have shown that x -+ Gh(x) oHx defines a continuous map N(n, 2) -+ COO(lR). The same is true for x -+ Gh(x), and each Gh(x) is a diffeomorphism. Since the diffeomorphisms of 34 lR form a topological group with respect to the subspace topology inherited from COO(lR), we conclude that N(n,2) --l- COO(lR), x I-? Hx = G,;}x) 0 (Gh(x) 0 Hx) is continuous. Now we get the result that the conjugacy h is smooth along all the I-dimensional foliations that are invariant foliations for the action of the group A. Recall that the foliations are strong, stable foliations for some Anosov elements, so they are Holder foliations (see, for example, [B-P]). An application of Journe's theorem [J] (see also [K-L2]), which is agam an identical argument as in [K-L2], implies that the conjugacy h is smooth. 35 6. Deformation Rigidity of Group Actions 6.1 Topological and Smooth Deformation Rigidity of Diagonal-block SL(n,Z) Action. We will first mention that SL(n,Z) action on N(n,k) for n"2: 3, k S n - 1 is topologically deformation-rigid. This follows from a general result in [Hu3]. This, together with a general philosophy that a topological conjugacy for a large group should be a smooth conjugacy, support the conjecture that S L( n, Z) action on N(n, k) is smoothly deformation-rigid. In this writing, we cannot confirm it for k "2: 3. The smooth deformation rigidity for k = 2, n "2: 3, n =I 4 is a corollary of Theorem A. First let us gIve some definitions that first appeared in [Hu3]. Let r be a finitely generated group, X a compact Riemannian manifold without boundary and <P : r x X --* X a smooth action. A deformation of action <P is a continuous 1-parameter family of smooth actions {<Pt : r x X --* X 10 s t S 1}, so that <Po = <p. Fix a set of generators {61 , ... , 6d } of r. For € > 0, an €-perturbation of <P is an action <PI : r x X --* X such that for each generator 6i , the diffeomorphism <PI ( 6i) of X is €-cIose to <p(6i ). So for a fixed deformation of ¢ and sufficiently small t > 0, <Pt gives an €-:perturbation of <p. DEFINITION 6.1.1. An action ¢ is topologically deformation-rigid if for any deformation {¢d of the action ¢, there exists an E > 0, and a continuous 1parameter family of homeomorphisms H t : X --* X, such that Ho = Idx 36 for any I E r and 0 ::; t < €. In case that H t are diffeomorphisms, we say the action </> is smoothly deformation-rigid or simply deformation-rigid. The approach to attack the deformation rigidity utilizes the theory of Stowe [St), where a criterion is given for the persistence of a fixed point of a group action. Let r act on a manifold M (denote the action by a), and p E M be a fixed point for the action. Let ao be the induced linear action on TpM, the tangent space to M at p. We denote by HI (f, TpM) the ordinary group cohomology with coefficients in this representation. PROPOSITION 6.1.2. If HI(f, TpM) = 0, then p is stable under perturbation of a; i.e., given any neighborhood U of p E M, there exists a neighborhood V of a E R(f,DifF(M)), such that each f3 E V has a fixed point in U. In light ofthis result, and also given that our SL(n, Z) action has dense periodic points, if the dense set of periodic points persists, then the topological conjugacy is defined by sending periodic points to the perturbed periodic points. To carry out this idea, we need several definitions [Hu3], and also a cohomology-vanishing theorem of Margulis [M1]. DEFINITION 6.1.3. Let E be a finite dimensional, real vector space, f a finitely generated group. A representation p : f -+ GL(E) is infinitesimally rigid if (1) the linear action of per) on E has 0 as the unique fixed point; (2) the first cohomology group of r with coefficients in the r -module E trivial; i.e., HI(r,Ep) = O. A representation p : r -+ GL(E) is strongly infinitesimally rigid if (1) the linear action of per) on E is hyperbolic for some I E r; 1S 37 (2) for all E-perturbation Ii of action p, the :first cohomology group of r with coefficients in the r-module E is trivial; i.e., HI(r,Ep) = O. The strong infinitesimally rigid action is clearly stable under small perturbation and the perturbed action is also infinitesimally rigid. DEFINITION 6.1.4. An action </> ofr on X is (strongly) infinitesimally rigid at a periodic point x E A if the isotropy representation is (strongly) infinitesimally rigid. The following theorem of Hurder gives a criterion for the topological rigidity of a group action. THEOREM (HURDER [Hu3)) 6.1.5. Let 4> be an Anosov action such that (1) the periodic points A are dense in X; (2) 4> is strongly infinitesimally rigid at each periodic point x E A. Then 4> is topologically deformation-rigid. From this theorem, the proof of the topological rigidity of our SL(n, IE) action reduces to the proof of the density of periodic points and the strongly infinitesimally rigid at each periodic point x E A. The first is the consequence that all rational points are periodic points (see Lemma 2.2.2), and the second is the consequence of a powerful theorem of Margulis [M1J, see also [Hu3]. THEOREM (MARGULIS) 6.1.6. Let reG be an irreducible lattice in a con- nected, semisimple algebraic R -group of higher rank, G, and assume that G~ has no compact factors. Then HI (r, lR~) = 0 for every representation p : r ~ GL(N, lR), N > 1. Combining the results above and those in Section 5, we obtain 38 THEOREM 6.1.7. Let r = SL(n,71) or any subgroup of finite index. Then tbe action of r on N(n, k) is topologically deformation-rigid for k ~ n - 1, n ~ 3. H k = 2, tbe action is smootbly deformation-rigid for n = 3,and =1= 4. 6.2 Deformation Rigidity of Aut(n, 2) Action. We now consider the whole automorphism group action on N(n, 2). Recall that a criterion of Hurder (Theorem 6.1.5) reduces the question to the calculation of the first group cohomology. vVe therefore introduce some easy facts about the first group cohomology. LEMMA 6.2.1. Let P : r --t GL(N, R) be a homomorphism, r 1 be a finitely- indexed subgroup ofr, and H 1 (r 1 ,plrJ = O. Let p(r) be generated by Anosov automorphisms p(rI), ... , p(rk). Tben Hl(r, p) = o. PROOF. Let 4> : r --t Rn be a cocycle; then flrl is a coboundary by the assumption; i.e., there exists an element v ERN, such that for all , E r 1, f(,) = v - p(r)v. Let ,f E r 1. Since f is a cocycle, we have Because ,f E r 1, we have f(,n = v - p(,nv = (I - p("{i))(1 + p("'(i) + ... + p(rf-l))V, and also since det(I - p(rf)) =1= 0, 39 we obtain det(I + pC--n) + ... + pClf- 1 )) #- o. Therefore, Now for any , E r, , can be expressed as , = ,i 'i l •• • q• Since f is a cocycle, we obtain that fCl) = v - p(,)v. o It is clear that this Lemma can be strengthened; e.g., we may assume that per) is generated by r 1 together with some matrices that do not have eigenvalues of roots of unity. But we are satisfied with the weaker form of Lemma 6.2.1. We define rm = {, E r:, = I(mod m)} for any group r c GL(N,JR). We have the following result. LEMMA 6.2.2. Let r = Aut(n,2), BSL(n, Z) the diagonal copy of SL(n, Z). Then rm = NmBSL(n,Z)m, where N = PROOF. It is a straightforward calculation. COROLLARY 6.2.3. rm It is obvious. 0 = Aut(n,2)m is generated by (m~ij ~) and BSL(n,Z)m. PROOF. {(fvt ~): M is the integer matriX}. 0 40 Let r = Aut(n, 2). Then rm is generated by Anosov elements. LEMMA 6.2.4. PROOF. We first prove that B5L(n,Z)m is generated by Anosov elements by an argument similar to Theorem 3.4; then our result follows easily. 0 In the next several Lemmas, we will prove that the first cohomology of Aut( n, 2)m is trivial for each m. The approach is simple. We first show that the whole group Aut(n,2)m is generated (up to the finite index) by several subgroups with vanishing cohomologies; then we use these data to show that the first cohomology of the group Aut( n, 2)m itself vanishes. For the sake of transparency, we first prove the statement for n = 3. We ,vill assume that r = Aut(3, 2) for Lemma 6.2.5-Lemma 6.2.9. LEMMA 6.2.5. 0)I 5L(3, Z)m mEn r m 2 C r(m,m2) :=< (m~l1 -1 0) (m~12 I 5L(3, Z)m mE12 ~), 0) (m~21 I 5L(3, Z)m mE ~) , 0)-1 ( (m~23 I 5L(3, Z)m mE23 ~), 0)-1 ( (m~32 I 5L(3, Z)m mE32 ~), 0)-1 ( (m~33 I 5L(3, Z)m mE33 ~) >c -1 ( I -1 ( I ( I 2l I I I PROOF. Take A will be = (! ~ ~); °°1 then the induced automorphism A' on [V, V] (~ ~ ~). A straightforward calculation gives °m rm. 1 0I)-I(AO A'0)( mEI12 41 Similarly, we may get (-m;Eij able E = (m~st ~) in this way by changing A, and using suit- ~) that appeared in the formula: GI ~). = (m~ll To get ( I -m 2 E12 n, we use A = To get ( I -m 2 E13 n,weuseA=G ~ n'E=(m~33 To get ( I To get ( To get ( To get ( m2E21 E 1 0~ 0)~ ~), we use A = (~ n,weuseA= G! ~). (m~23 ~ ); ,E = (m~23 o m) (I o mE E= I -m 2E23 ~), we use A = -m E31 ~ ~), we use A = (~ o~ ~) I - m2E32 n,weuseA= G! D'E=(m~23 G 1 0 ,E = 1 1 . ' E _ ( - 21 I mEn ~ ~), E = ( I 1 mE32 m PROOF. o Since r m2 C r(m,m2) C r m, it is clear that r(m,m2) is a finitely indexed subgroup of r m. Recall that r m is generated by Anosov elements; our result then follows. 0 Next we want to prove that H1(r(m,m2),R6) = o. We need some facts about the intersections of those copies of BSL(3, 2)m with BSL(3 , 2)m itself. 42 (~ ~ {), then the induced action on [JR',JR'] is LEMMA 6.2.7. H A = A' = PROOF. af-de ai - cg di - fg It is a straightforward calculation. LEMMA 6.2.8. bf-Ce) bi - ch . ei - fh 0 For all integers b, c, d, f, g, h, 0 0) (I 0)-1 BSL(3,Z)m ( I 001 1 ( ae-bd ah - bg ( dh - eg md 1 mf EEl E mIl mu ~) n BSL(3, Z)m; 0)-1 0)I n BSL(3, Z)m; I BSL(3, Z)m (I 0)I BSL(3, Z)m (I 0)I n BSL(3, Z)m. mE 32 -1 PROOF. mE33 Let us prove the first formula. 1,) BSL(3, 0)-1 (A 0) (I 0) (-mEllAA+ mA' Ell lOA' mE ll I - Let A E SL(3, Z); then (~ E Z). So Therefore, I 0)-1 (A '0) (I 0) BSL(3,Z) lO mEn I (mEu E 43 if and only if -mEllA + mA'Ell = O. This condition forces 0) be) ( ae - bd a a a a = ah - bg a a . ( a a a dh - eg a a a Then what is left is a straightforward calculation. 0 Having had these Lemmas in hand, we are in a position to prove the vanishing PROOF. Let I : r(m,m2) IIBSL(3, Z) as well as II -t R6 be a cocycle. (m~ij ~) -1 By a theorem of Margulis, BSL(3, Z) (m~ij ~) axe cobound- arIes. Let IIBSL(3,Z)C,) = v - PC,)v, II (m~ij ~) -1 BSL(3, Z) (m~ij ~) C,) = Vij - PC,)Vij. We want to prove that Vij = v for all i,j. For example, let us show that VII = v. In fact, id,f = E (~da ~a ~/) 1 I ( mEn 0) BSL(3, I -1 ( Z)m I mE l l ~) nBSL(3,Z)m; 44 hence, v - p('Yd,!)V = V11 - P("(d,j )V11' or v - v11 = P("(d,!)( v - V11). This is to say that (v - V11) is a common eigenvalue for all p( 'Yd,!) with d, fEZ corresponding to eigenvalue +1. Let _ (ll) v - V11 - WI (ll) 'W 2 (11) 'W 3 (11) 'W 4 (11) 'W 5 (l1»)t. , 'W 6 notice that (ll) 1 md ° ° °° °° °° °° °° ° ° ° ° °° °° °° °° °° P("(d,j)( V - Vll) = 1 we have that (11) w2 mf 1 1 mf 1 md w3 (11) w4 (11) w5 1 (11) W6 (11) WI (11) (ll) + m d WI(ll) + w 2 m f w3(11) w2 (11) (11) w3 w3 (11) w4 w 4(11) + m f W5(11) (ll) (11) W5 d (11) + (11) m w5 w6 . (ll) (ll) W5 (11) w6 (ll) = w3 = w5 = 0. Therefore, (ll) v - v11 = ( 0,w 2 (11) ,0,w4 (l1»)t . ,0,w6 Similar consideration yields °° V - v12 = ( WI12 , , 'W 4(12) 'W 5(12) ,0 )t v - V21 = ( 0,0'W 3 (21) (23) v - V23 = ( WI (23) ,0,0, W4 (32) V - V32 = ( 0,0'W 3 (33) v - V33 = ( 0'W 2 (21) ,0'W 5 (32) 'W 4 (33) ,0'W 4 (21»)t 'W 6 (23»)t , W5 (32) 'W 5 w2 (11) (11) WI for all f, dE Zj It forces WI (11) WI (11) WI ,0 )t (33»)t ,0'W6 . (11) - w3 (11) w4 (11) W5 (11) w6 45 Let us then show that w~ = 0 for all i,j, k = 1,2,3. For example, wp = o. Consider G1 : = ( n( m EI 0)I BSL(3, Z)m mE 0)I BSL(3, Z)m mE -1 11 -1 I mE12 I ( 11 I ( 12 a straightforward calculation shows that ( -mE 12 B ~ mB' E12 ~,) E G 1 with B= and B'= 1 + ma ( md o mb l+me 0 me) mf 1 0 (1 + ma)(mf) - (md)(me) l+ma md (mb)(mf) ) mb , l+me such that the following conditions are satisfied: (1 + ma)(l + me) - (mb)(md) = 1 1 +ma - md =1 1 + me - mc = 1, f = c. (For example, B = 1 + mk ( ~k -mk 1 -Omk ml) ~l ). Given this fact, together with a similar argument for the assertion of the possible forms of v - Vij, we conclude that and ll W 2 -- w 6ll -- w 112 -- w 512 -- 0 . 46 To conclude that Wil = 0, we may consider G2 : = ( n( m EI 0)-1 ( I BSL(3, Z)m mEn ~) 0)I BSL(3, Z)m mE ~) I 11 -1 I mE21 ( I 21 and do the same analysis. A similar argument can be used to prove that w~ = 0 for other i,j, k E Z. This proves that when f is restricted to the specified copies of BSL(3, 2), we get f(-y) = v - p(-y)v. But our original group is generated by these copies; this proves that f is a cocycle. 0 To prove the vanishing for general n ~ 3 with the coefficient in the standard representation Po, the argument we had goes through. So we have PROPOSITION 6.2.10. Hl(r m, ]RN) = 0 for r = Aut(N(2, n)), dim(N(2, n)) = N. To prove that the action po is strongly infinitesimally rigid, we need to prove that the first cohomologies are also vanishing for the coefficient in the perturbed representation. 0 hence HI (r m, Pt) PROPOSITION 6.2.11. H1(r(m,m2), Pt) - o for suffi- ciently small t and n ~ 3, n =1= 4. PROOF. Recall that the diagonal-block action is locally rigid. Hence for sufficiently small t, PtIBSL(n,Z) = h-;1 Poht with h t smoothly close to identity. It is clear that other copies of SL(n, Z) also act on the manifold in a locally rigid way, so ptl ( ~) = hi;;' po hi;' with hi;, smoothly close 0) (I BSL(n,Z) mEij mE -- I I to identity. -1 '1 We may go over the proof of Proposition 6.2.9 again with necessary modification, which gives the proof of this proposition. 0 47 Now we are ready to prove the deformation rigidity for the Aut( n, 2) action on the nilmanifold. THEOREM 6.2.12. Aut(n,2) actions on nilmanifold N(n,2) are smootbly deformation-rigid if n = 3, n = 5,6, .... PROOF. The proof of this theorem uses a criterion of [Hu3]. See Theorem 6.1.5. The only thing needing proof is that Aut(n,2)x (which is the subgroup of the Aut(n, 2) fixing point x E N(n,2)) contains Aut(n, 2)m for x, a rational point, and for some integer m. But it is clear. 0 Remark 6.2.13. It is easy to show that for any finitely indexed subgroup f c Aut(n,2), the action of f on N(n,2) (with n 2:: 3, n =I 4) is also smoothly deformation-rigid. We will sketch the argument below. (1) f n BSL(n,Z) is a finitely indexed subgroup of BSL(n,Z). Indeed, for all A E BSL(n, Z), Ad E f for d =the index. (2) The same reason implies that f n UN(UN { (~ ~)}) is a finitely indexed subgroup of UN. So we will have the following. (3) There exists an m such that BSL(n, Z)m C f. ( 4) There exists an m' such that UNm' C f. Therefore, we have Aut(n,2)mm' C f. The computation of cohomologies goes through. So we will have smooth deformation rigidity. 0 48 7. New Examples of Cartan Actions and Other Anosov Actions on Tori and Limitation of Our Method In this section, we will give a complete list of linear SL(n, Z) actions on tori for fixed n. Among them, Cartan actions and Anosov actions are specified; also, the actions with "complete I-dimensional splitting foliations" are singled out. Using the local rigidity theorem for Cartan actions in [H-K-L-Z], we then have a number of locally rigid SL(n, Z) actions on tori. These examples are new; the construction is natural and simple. V. Nitica informed me of the classical results about Young tableaux and a recent survey. I thank him for his generous help. We remark that the examples of higher-rank actions on tori appearing in the literature are SL(n, Z) actions on 'lI'n (locally rigid if n ~ 3), Sp(n, Z) actions on 'lI'2n (locally rigid if n ~ 2), and also Hurder's examples using the trick of A. Weil, i.e., "restriction of scales." The latter examples can be described as follows. Take an algebraic number field k of degree dover Q; let O(k) be the ring of integers for the field and let SL(n,O(k» be the subgroup of SL(n, k) with the entries from O(k). Then take any r c SL(n, O(k»; there exists an analytic "standard" action of r on 'lI'dn. In our list, the Cartan actions are those obtained from exterior tensor product representations; the Anosov actions are those obtained from decomposing the kfold tensor product representations into irreducible ones with k =I O( modn); the actions with "complete I-dimensional splitting foliations" are those obtained from symmetric tensor representations. Let us first look at some examples. Instead of considering the automorphisms on free k-step nilpotent algebra (we remark that the construction of automorphisms of the free, k-step, nilpotent Lie algebra also gives the automorphisms of vectorspace automorphisms and hence may factor through to toral automorphisms) we 49 consider the automorphisms on the exterior algebra E( V) of a real vector space V. We first quote some basic facts from the theory of exterior algebra. (See D. G. Northcott: Multilinear Algebra [N] for a detailed treatment. For our purpose we will assume all modules in [N] to be real vector spaces). Let f : M -+ N be a linear map between vector spaces .lvI, N, let E( M), E( N) be the exterior algebras. Then f has a unique extension to a homomorphism E(f) = ffiEp(f) of graded algebra E(M) = ffiEp(M) to E(N) = ffiEp(N). Moreover, if ml, ... ,mp EM, then If 9 : K -+ M is another linear map between real vector spaces 1\,lv[, then E(f) 0 E(g) = E(f 0 g) It is clear then that if M = N = K := V and f : V -+ V is an isomorphism, then E(f) as well as Ep(f) is an isomorphisms. In other words, any matrix subgroup of GL(n, JR) acts on E(V) as well as Ep(V). Let V be an n-dimensional vector space; then we may consider E(V) (Ep(V)) as a real vector space. Fix a basis {el, ... , en} of Vj then {eil (1 ~ i l ~ n)j ei 1 / \ ei 2 (1 ~ i l < i2 ~ n)j ... jei1 /\ ... /\ ei n (1 ~ i l < ... < in ~ n)} is a basis for E(V), and {eil /\ ... /\ eip (1 ~ i l < ...... < ip ~ n)} is a basis of Ep(V). Let us take a lattice generated by the integer points with respect to the chosen basis. Then any subgroup r of S L( n, Z) induces an action preserving the lattice. So it factors through a torus action. Example 7.1. Let r c SL(n, Z) be a subgroup of finite index with n 2: 3; then it acts on 1I'C;) , 1 ~ p ~ n - 1 through the action of ron Ep(V). The action is a Cartan action and is locally rigid. 0 50 It is worth mentioning that the actions constructed above and also examples 7.5 are actually irreducible by an easy argument, using the well-known Schur's Lemma. On the other hand, we cannot say anything about the irreducibility of the action constructed in the same manner, using automorphisms of free, k-step, nipotent Lie algebra. Example 7.2. We may group certain actions above together; in other words, we may consider the actions r on the spaces of the form EBiESEi(V) factors a lattice, where SCI = {1, 2, ... , n -1}. The actions then are Cartan actions. The actions are also rigid if n 2:: 3. 0 It looks as if the actions here are product actions, but they are not. They are "diagonal actions" of some product actions. For Examples 7.1 and 7.2, we have enough 1-dimensional foliations for an Abelian subgroup to trellis the tori, and each of the foliations is the intersection of the stable foliations of several Anosov elements, and these foliations are also the strongest foliations for some Anosov elements in that Abelian subgroup. We therefore can use the arguments in [H-K-L-Z] and prove that the actions are locally rigid. If we consider other tensor algebras, for example, tensor algebra T(V), Tp(V) or symmetric tensor algebra S(V), Sp(V), the latter having been already introduced in Section 5, we may construct other examples of actions on tori. Example 7.3. Let r C SL(n, Z) be a subgroup of finite index with n 2:: 3; then it acts on Tnp,p =1= O(modn), through the action of r on Tp(V). The actions are Anosov actions and do not have enough I-dimensional foliations for a suitable Abelian subgroup in r. 0 Example 7.4. We may group certain actions above together; in other words, we may consider the actions r on the spaces of the form EBiESTi(V) factors a 51 lattice, where SCI = {k E Z+; k =/:. O(modn)}. The actions then are Anosov actions. 0 Example 7.5. Let r C SL(n, Z) be a subgroup of finite index with n 2: 3; then it acts on Td(p),p =/:. O(modn), d(p) = (n+;-l) through the action of r on Sp(V). The actions are Anosov actions and do have enough 1-dimensional foliations for a suitable Abelian subgroup in r to trellis the tori, but some of the foliations can never be realized as strong stable foliations for some Anosov elements in r. Nevertheless, the foliations are intersections of stable foliations for some Anosov elements and therefore persist under small perturbation. 0 Example 7.6. We may group certain actions above together; in other words, we may consider the actions r on the spaces of the form EBiESSi(V) factors a lattice, where SCI = {k E Z; k =1= O(modn)}, kl =1= k2 (mod n) for all kl' k2 E S, and S is finite. The actions then are Anosov actions if n 2: 3. The actions have enough 1-dimensional foliations for a suitable Abelian subgroup in r to trellis the tori, and the foliations are intersections of stable foliations for some Anosov elements, and therefore persist under small perturbation. 0 We cannot use the Cartan action theory to attack the rigidity of the actions in example 7.3-7.6. But it is clear that the possessing and persistence of enough 1dimensional foliations in example 7.5, 7.6 make it possible to generalize the Cartan action theory to get the rigidity for these cases. It is a classic result that all irreducible, rational representations of S L( n, Z) on finite-dimensional vector spaces are polynomial representations, and they are obtained by decomposing the representation in Example 7.3 into irreducible representations. In fact, we haye the following parameterization for polynomial representations by Young tableaux [Sun]. We will omit the definitions of Young tableaux and their properties. 52 We denote by A a Young tableau; we let P>. be the irreducible polynomial representation corresponding to it. We say that the length of A is p if P>' is contained in SL(n, Z) x Tp(V) -+ Tp(V). PROPOSITION 7.7. Fix a Young tableau A; we have the following: (1) PX preserves integer points with respect to a "standard basis," and hence factors through a toral automorphism also denoted by P>..; (2) the toral action P>. is Cartan iff P>. is one of the actions listed in Example 7.1; (3) the toral action p>.. is Anosov iff the length of A is not equal to O(modn); (4) the toral action p>.. has 1-dimensional splitting foliations iff P>' is one of the actions listed in Example 7.1 and in Example 7.3. The proof of this proposition is easy. We omit the proof. Next we will give a description of all the linear toral SL(n, Z) actions. PROPOSITION 7.8. Any linear SL(n, Z) toral action on ']I'N induces a homo- morphism between SL(n, Z) and SL(N, Z). For n ;::: 3, the homomorphism is a polynomial homomorphism restricted to a subgroup of finite index. This is proved in [Ste]. The two propositions above describe all the linear SL(n, Z) toral actions in terms of Cartan action, Anosov action and action with "complete I-dimensional splitting foliations." Although we may have a number of other actions on tori, for example, we consider V with dimension 2n; we consider Sp(n, Z) on the factored tori; we cannot get the rigidity results using the Cartan action theory. vVe hope we may prove the rigidity result for this class of actions using another approach that is well-developed in [K-L]. But we also have to exclude those actions with length O( modn) of the Young tableau. 53 We would like to ask the following question. Fix an integer n ~ 3. 'What is the torus of the biggest dimension on which SL(n, Z) could act locally rigidly? If, by any chance, that action is algebraic, we have one of the above examples as an answer. So it is clear that either the torus is obtained as in Example 7.2 or such a torus does not exist. It is worth mentioning that our method of using the Cartan action theory to prove rigidity is impossible to be carried out in the case of the nilmanifolds N( n, k) of higher steps, i.e., k ~ 3. The reason is simple: We do not have an Anosov element at all for k ~ n, and also we do not have enough I-dimensional foliations for a suitable Abelian subgroup to trellis the space. This is the limitation of the method we use. Some progress has been made by A. Katok that for certain cases the density of the orbit of the abelian group in the higher-dimensional foliation is enough to prove the rigidity. But this is not the case for our nilmanifolds N(n, k) with k ~ 3, because the orbit is never dense. We want to state a conjecture to finish the first part of our paper. Conjecture. (1) The SL(n, Z) action on N(n, k) is locally rigid if n ~ 3, k::; n-l. (2) The group Aut(n, 2) of automorphisms of N(n, 2) is locally rigid. We remark that our results in the paper support this conjecture. 54 PART II: Anosov En Actions 1. Introduction Hyperbolic behavior in dynamical systems has been investigated extensively, especially for Anosov diffeomorphisms and Anosov flows. See, for example, [A] [PI] and references there. Anosov diffeomorphism on a compact manifold possesses nice structures and properties. It foliates the manifold by stable foliation and unstable foliation. It is structurally stable in the sense that any perturbation of the Anosov diffeomorphism is still an Anosov diffeomorphism, and they are topologically conjugate. It is ergodic provided that it preserves a volume. And it has two naturally associated measures, the SBR measure and the Margulis measure. For these two measures, the diffeomorphism is just a Bounoulli shift and hence ergodic, weakly mixing and strongly mixing. For the Anosov flow on a compact manifold, the development is parallel to the Anosov diffeomorphism case. We have the same properties for the Anosov flows as those for Anosov diffeomorphisms with minor modifications. Two generalizations of the Anosov diffeomorphisms and flows have been made; one is called normally hyperbolic dynamical systems; the other is called Anosov group actions. We call a group action Anosov if the group contains at least one Anosovelement [P-Sl]. In our work, we are interested in a special kind of Anosov group actions, namely, the Anosov En actions. We now give the definition and some examples of Anosov En actions. Let M be a compact manifold without boundary. Let cP : En X M --+ A1 be a smooth map such that CPt: M ~ M, x.- cp(t,x) is a diffeomorphism. We call cP an En action if 7r : ]Rn --+Diff(M), t.- CPt is a group homomorphism. 55 In this work, we will assume that 7r is locally free; i.e., for any x E M, there exists a neighborhood U of origin in R n, such that 7r( u)x =1= x for all u E U - {o}. It is clear that the orbits of the locally free Rn action define a smooth foliation nn. DEFINITION 1.1. The locally free R n action <I> is called an Anosov R n action if there exists an Anosov element; i.e., there exists an element r E Rn such that f = <Pr : M --t M is hyperbolic at nn, or in other words, T f : M --t M leaves invariant a splitting contracting ES more sharply than Tnn, expanding EU more sharply than Tnn. EXAMPLES 1.2. The following are a list of currently known examples of Anosov R n actions. (1) The standard actions of IRn on Tn; (2) Anosov flows; (3) Suspensions of Zk Anosov actions (Zk Anosov actions are actions generated by Anosov diffeomorphisms); i.e., let the action of Zk on N be anosov; let Zk be embedded in IRk as a lattice. Consider the action of Zk on IRk x N by z(x, n) = (x - z, zn) and form the quotient Note that the action of R k on IRk X N by r( x, n) = (r + x, n) commutes with the Zk-action and therefore descends to an JRk-action on M. (4) Weyl chamber flows; i.e., let G be a real connected semisimple Lie group of the non-compact type, and r be a cocompact lattice, A being a splitting Cartan subgroup. Recall that the centralizer of A splits as a product !vI A, where M is 56 a compact group. It is clear then that A acts on M\Gjr from the left; it is an Anosov action [I]; (5) Assume the notations of the last example. Let p : r -+ SL(n, Z) be a representation of r. Suppose that r acts on a compact manifold N via p such that A action on N via p is Anosov (e.g., see § 7 of the first part of my thesis). Then r acts on M\G x N via ,(x, i) = (x,-l"i). Let V := M\G Xr N := (M\G x N)jr be the quotient of this action. As the action of A on M\G x N given by a(x, i) = (ax, i) commutes with the r-action, it induces an Anosov action of A on V. (6) Suspension of the above]Rk actions; i.e., let <P : ]Rk X M -+ M be an Anosov ]Rk action on M, let Z =< 91, ... ,9n > act on M, such that 9i commutes with the ]Rk action (in other words, 9i<Pt = <Pt9i for all i E ]Rn). Let zn be embedded in ]Rn. Let Z act on M x ]Rn via z(m,r) = (zm,z - r). Let V = (M x ]Rn)jzn be the quotient space. Then the action of]Rn on M x ]Rn VIa (s, i)(m, r) = (sm, i + r) commutes with zn action on M x ]Rn and hence descends to an ]Rn+k action. It is clear that this action is an Anosov action. (7) Any product actions of the above. D The definition of Anosov actions first appeared in [P-S1] in a more general form in 1972. In that paper, ergodicity of the action was considered. In [P-S2], they also proved that for an ergodic group action, almost every element in the group is ergodic. 57 In my work, we will consider the individual elements of the group lR. n and prove that every Anosov element is ergodic. The unsolved interesting problem is whether or not the Anosov elements form a dense set. We remark that although there are plenty of examples of Anosov diffeomorphisms and Anosov flows (for the Anosov flows, even non-algebraic examples have been constructed), there are relatively few examples for the higher-rank nonproduct Anosov Rn-actions. Conjecturally, there are "too many" structures inside such actions that inhibit the non-trivial perturbations. On the based of this observation, A. Katok, R. Spartzier conjectured that such actions should be locally rigid. This is confirmed in a recent paper [K-SJ for Weyl chamber flows, suspensions of Anosov Zk-actions on tori, and "twisted Weyl chamber flows" (Example 1.2(5) with N = 'JI'k). This work is inspired by a talk of R. Spartzier at Penn State in March 1991. The examples listed above are due to R. Spartzier and A. Katok. Our work is an attempt to understand the dynamics of Anosov lR. n actions, reveal the rich structures of the actions, and it is hoped, to give additions to the rigidity investigation. 58 2. Suspension Construction of Anosov ]Rn Actions Let JR n x M ~ M be a volume-preserving Anosov action. We call an element in JRn a regular element if it is an Anosov element. Similarly, any I-parameter subgroup containing a regular element (hence every non-trivial element is regular) is called regular. Fix a regular element r E JR n and let j = «P r . Let strong stable and strong unstable manifolds at x, W: s , W: u be W:, W: be stable and unstable manifolds for j at x. We will later abuse the notation not to distingush rand q>r. LEMMA 2.1. SS g W x = Wss gx, uu gWxUU = W gx . (2) IT x, y are in the same JRn orbit, then W xSS = gWySS , for some 9 E JRn. PROOF. (1) Since W;: = {u EM: lim d(fnu,jngx) = O} n-oo But g is a diffeomorphism, so the above is 59 So (1) is true. (2) is a corollary of (1). 0 We will call x E M a periodic point if lR nx is compact. We denote the set of all the periodic points by n. Then Burns and Spatzier [B-S] proved that n = M. We recall a result that is called the "Product Neighborhood Theorem." We fix a metric d on M. Let du , ds, d uu , d ss be the induced metrics on the leaves of foliations :Fu,:Fs , :Ft&'U, :Fss , respectively. We define for x E M,8 > 0 Bx(8) = {y EM: d(x,y) < 8} B;(8) = {y E W:(8) : du(x,y) < 8} B!(8) = {y E W:(8) : ds(x,y) < 8} B;U(8) = {y E W: U(8) : duu(x, y) < 8} B!S(8) = {y E W: S(8) : dss(x, y) < 8}. Then, there exists 80 > 0 independent of x E M such that for 8 :::; 80 the maps given by G(y, z) = B:(28) n B;U(28) H(y, z) = B:S(28) n B;(28) are unambiguously defined and are homeomorphisms onto their respective images, which are called product neighborhoods of x. This result is proved by Hirsch, Pugh and Shub for the flow case in [H-P-S] and is a straightforward generalization for general Anosov actions. This result is used in [P-S1]. 60 LEMMA 2.2. Wux = M , Wxs = M for all x E M. PROOF. If we can prove that W~ is open, then from the fact that M is connected, we are done. Let z E W~. Take a product neighborhood N of z. Take pEN, a periodic point for the action. We may choose so that p is a periodic for a one-parameter subgroup gt, which is a regular one-parameter subgroup of Rn, and it has the same stable, strong stable, unstable, strong unstable manifolds at any points x E M as f has. Let gtoP = P for some to E R+. Now W;s intersects W; at q E W;, so it will intersect W: at q' E W:, so we have · gntoq , = p, and 9ntoP = p, 11m n--oo but gntoq' E W:, so P E W~. 0 THEOREM 2.3. If W:u i= M for some x E M, then for all y E M, W;u i= M. PROOF. We first prove that if W:u i= M for a periodic point x E M. Then W;u i= M 'Vy E M; secondly, we prove that if W:u = M for all periodic points, then W;u = M 'Vy E M. Let K = W~u for x a periodic point in compact orbit CO(x)j then u yECO(x) W;u = U <PtW:u ~ W:, tE some compact set of JRtn the middle term here, is actually the image of a compact set under the continuous map so UyECO(x) W;u is compact, but it contains a dense set W:, so it is lvI. Now take any y E Mj let y E <ptlK for some t1 ERn; we claim W:u C <ptlK. Indeed, Lemma 2.1 implies that <p~lW:u C K So W:u C <PtJ(, so W;u i= M. 61 In the next several lemmas, we will prove that if W;L1L points x E M, then it is also true for all x E ]vI. LEMMA 2.4. M for all periodic 0 Let '"n\ = lRn/Zf, ... , 'fk = lRn/Z k be n-tori, Zi rv zn. Let lR n act on 'f 1 x··· X 'fk by translation on each factor. Then for any l-parameter subgroup gt oflR n , for any t: > 0, and for a dense set of (PI, ... ,Pk) E 'f 1 x ... X 'fk, there exists a sequence {td with ti --t 00, such that d(gtiPj,pj) < t:, for all j = 1,2, ... , k. Since every element of lR n acts on 'fIX' .. x 'f k by a volume-preserving PROOF. map, the result then follows from Poincare's recurrence theorem. LEMMA 0 2.5. Let W;u = M for all periodic points Pi then W;'U = M for all yEM. PROOF. Let x E M be any point in M, and Be(x) be a t:-neighborhood of x. We want to prove that any W;u intersects this neighborhood for any y E lvI. Take a finite.t:/2-net, the center of which, Pi, i = 1,2, ... ,k, are periodic points. Let 8 be small enough such that if we take another t: /2-net centering at qi with d(qi,Pi) < 8, it still covers .lvI. Let Pi E Gi, which is the compact orbit containing Pi. Using the fact that lR n acts transitively on Gi and that every orbit is an n- dimensional manifold, we get that Gi ~ 'f i = 1R n /Zi and the action is the usual translation. Now take a I-parameter subgroup gt containing f. We claim that there exists a to > 0, such that for a dense set of qi in a small neighborhood of Pi, gto(W~~~e/2) n B e/ 2(x) =f:. <p. Indeed, w;i~TnBe/2(X) =f:. <p for sufficiently large T, so there exists a small number c > 0, such that if It I < c, and qi sufficiently close to Pi gt(W~~T) n B e / 2 (x) =f:. <p. From Lemma 2.4, we know that for a dense set qi close to Pi and 62 to ~ T, such that d(gtoqi, qi) is very small and hence gto(W~~T)nB~/2(X) =1= </>. But it is clear that WgUU /2) if to is sufficiently large, so 9to(Wq~Ue/2) n . T C gto (W~U to q" q. ,€ ., B~/2(X) =1= </>. This claim tells us that we may find q E W~u n g-to(B~/2(X)) such that q E Wq~~e/2' Now g_to(W;U) intersects with some B e/ 2(qi) (this is because all B~/2(qi) cover M); take q E W~u n g-to(B~/2(X)) (the non-emptiness is proved in the above argument); let us assume that q E W~~e/2' Take r E g_to(W;U) n W;. It is clear from the fact that g_to(W;U) n B e/ 2(qi) =1= </>, that we may assume that d(q, r) < £/2, so But gto (r) E W;u, so we are done. 0 DEFINITION 2.6. An Anosov Rn action is called irreducible if W:u = M with respect to some regular element (hence for all regular elements) for some x E .VI (hence for all x EM). Otherwise, the action is called reducible. THEOREM 2.7. 0 HW:u =1= M for some x E M, tben tbere exists a compact set K C M sucb tbat (1) 3Rk C Rn sucb tbat RkK = K; (2) let Rn-k be a complement ofR k eRn; tben tbe action is tbe suspension of (K, R k) by n - k bomeomorpbisms in R n- k; (3) K is a C 1 submanifold of M. (Hence tbe bomeomorphisms in (2) are actually C 1 diffeomorpbisms.) PROOF. Since W:u =1= M, we may assume that for some periodic point p E -'1, K := Wpuu =1= M. (1) Let G = {g ERn: (Rg)K C K}. It is clear that G is a group and it has to be R k for some k. 63 (2) Let lR n - k be a complement of lRk. Let lR n - k = lRel EB lR C2 EB ... ffi 1Re n -k; it , y , n-k copie s is easy to see that for each ei, 3ti such that tieJ( = I<, and if -t i < t < ti, teiI< =I I<. Indeed, we denote tel by g~1); let s be the smallest, positive, real number such that I< n gF)(I<) =I 0. That such an s exists may be seen as follows. Suppose there exists arbitrarily small positive t E lR such that I< n g?)(I() =I 0. This implies that K = g~l)t(K) for arbitarily small t and all n E Z. This means that {t E lR : I< = g~I)(K)} is dense in lR. Thus, I< has to be invariant under gF), which is a contradiction. Without loss of generality, let eiK = K, tK =I K for 0 =I It I < 1. Then our action is obviously the lR k action on I< suspended (see §1 Example 6) by We use a fact here that <PtK, t E lR n foliates the manifold AI. We will prove this fact in the appendix. (3) The proof is basically the copy of Anosov's argument. vVe will produce the proof in the following Lemmas. D We remark that this theorem actually asserts that any Anosov action can be "decomposed" into the suspension of an irreducible C 1 action. Following Plante [PI], we call a set ScM F-saturated for a foliation F if it is a union of leaves of F. It is clear that that the closure and complement of an F-saturated set are again F-saturated. LEMMA 2.8 . Let FSs, FUu, Rk be tbe strong stable, strong unstable and lR k orbit foliations witb dimension m, I, k , respectively. Tben (1) K is Rk saturated; (2) K is FUU saturated; 64 (3) K is FSS saturated. PROOF. (1) and (2) are clear. Suppose that (3) is not true; let P E K and q E W;s, q 1:. K. Let q = <Pspo for Po E K. Then K n <PsK = <p. Take to to be a regular element close enough to J, such that <ptoK = K, <Pto<PsK = <PsK. Then <pntoK = K, <Pnto<PsK = <PsK. So lim n __ = d( <pntoq, <pntop) = 0, so two compact sets K, <PsK have distance 0, so they intersect. We get a contradiction. LEMMA 2.9. 0 Let Wo E M and WI E W!~. Consider H : p E W~o --t q E W~l by sliding p along W;s to hit W~l at q. Then H maps RkW;s to RkW;s. PROOF. Otherwise, if we choose WI close enough to wo, we know that there exists a small neighborhood of 0 in R n-k such that there exists tl in it, and <Pt1 K = K, which is a contradiction. 0 The foliations n} FUU FSS are said to be jointly integrable if there exists a C I foliation F such that dim F = m + I + k and any leaf of n k, F UU or FSS is entirely contained in the leaf of F. LEMMA 2.10. The foliations n k , FUU and FSS are jointly integrable. So K is a CI manifold. PROOF. We now use Anosov's argument. It is easy to see that K is C I . We are not going to repeat the argument. We refer the reader to [A]. 0 Appendix 2.11. We will give a proof that if for a periodic point p E M K = W;u =1= M, then ¢>d< form a partition of M. In other words, either K n <Ptf{ = <P or K = <PtK. Following Theorem 2.7, we let R k be the maximal subgroup in R n such that K is invariant under Rk. Choose n - k, regular I-parameter subgroups g~i) from 65 the complement of R k in R n, which generate R n-k, such that the stable, unstable, strong stable and strong unstable foliations are the same as those of <p, and p is a periodic point of g~i) with period rio It is clear that K cannot be invariant under any g~i). Let Ko C W;U(Ko =1= 0) be a minimal set with respect to the following conditions: (1) Ko is closed in Mj (2) Ko is Fuu-saturated and invariant under lR k j ( i) (3) gri (Ko) = Ko. Such a set exists by Zorn's Lemma. We claim that Ko, <PtKO either disjoint or coincide. Indeed, Ko n <PtKO satisfies (1),(2),(3), so our result follows. So it is then clear that <PsKo, <PtKO either disjoint or coincide. Remember also that UtElRn <PtKO = M because the set contains an unstable manifold and is also com- pact, so <PtKO = M, tERn is a partition of M. Because W;u is in <PtKO for some t E lR.n, this forces K = Ko. 0 66 3. Irreducible, Weakly Mixing and Continuous Spectrum In [P-S1], the ergodicity of a general measure preserving Anosov actions is established. Our effort here is to show that the structure of irreducible Anosov lR n actions are richer than being ergodic. It is actually weakly mixing, which is equivalent to the fact that the induced unitary representation does not have measurable eigenfunctions except the constant eigenfunctions. Moreover, the action is metrically transitive. The latter will be proved in the next section. We first recall some definitions. DEFINITION 3.1. A lR n action on M is ergodic iff it is measure-preserving and all measurable, invariant functions are constant; it is weakly mixing iff lim r ..... oo m t (~()) [ I(U j, g) - (j, l)(g, l)ldt = 0 r ) B(r) for all integrable functions j, gj it has continuous spectrum iff it has no measurable eigenfunctions other than the constants. 0 In this section, we will prove the equivalence of the concepts of irreducibility, weakly mixing and continuous spectrum. THEOREM 3.2. The following statements are equivalent: (1) Anosov lR n action is weakly mixing: (2) the action is irreducible; (3) the action has continuous spectrum. o We will first prove the equivalence of (1) and (3). LEMMA 3.3. Let j be a bounded, measurable function. Then lim r-oo m (~( r )) J[B(r) If(t)ldt = 0 67 . a su bset J C l!]) 1'ffth ere eXlsts ~ n, l'Im r _ oo m(JnBr) m(Br) = 0 suebtl, la t lim IJ(t)1 = O. Itl-oo t~J Hence, lim (~()) r If(t)ldt = 0 r } B(r) lim C~C r )) }rB(r) IJCt)1 2 dt = o. r-oo m iff r-+oo m PROOF. The argument is standard. We first prove that sufficiency. For any c > 0, take ro > 0, such that if It I ~ ro, t tt. J, IJ(t)1 < c/3. Now mc~(r)) her) IJCt)ldt = m(~(r)) IJCt)ldt + m(~(r)) lIJ(t)ldt l(r)-J < 1 r - m(B(r)) JB(ro)-J IJ(t)ldt + 1 r m(B(r)) JB(r)-B(ro)-J IJ(t)ldt + m(~(r)) l'J(t)'dt ~ c/3 + m C~C r )) JrB(r)-B(ro)-J IJCt)ldt + m C~C r )) JrJ IJ(t)ldt. If we take r sufficiently large, we get the last two terms ~ c/3 + c/3. So we proved that lim r-oo m (~( r )) }rB(r) If(t)ldt = o. Next we prove the necessary part. Let Jk = {t ERn, IJ(t)1 ~ t}; then it is clear that J 1 c he··· c Jk+1 C .... We also have r 1 1 1 m(B(n)) JB(n) IJ(t)ldt ~ m(B(n)) km(h n B(n)), h c 68 · n --..oo m(JknB(n» so IIm m(B(n» = 0 . S0 we may fi n d a sequence such that m(JknB(n» m(B(n» < k1 £or a 11 n > _ 1k. Let J = U~l[Jk+l n (B(lk+d - B(lk))]j then if lk ::; n < lk + 1, we will have J n B(n) = (J n B(lk)) U (J n B(n) - B(lk)) C (Jk n B(lk)) U (Jk+l n B(n)). So m(J n B(n)) m(B(n)) < m(Jk n B(n)) + m(Jk+l n B(n)) - m(B(n)) 1 1 m(B(n)) ::;k+k+l --+ o. Now we show that limt --.. oo IJ(t)1 = o. Indeed, we removed from B(lt) those t's up such that IJ(t)1 > Ij we removed from B(12) - B(lt) those t's such that IJ(t)1 > ~j we removed from B(lk+l) - B(lk) those t's such that IJ(t)1 > ktl j hence it is clear that if lk ::; It I < lk+I, then IJ(t)1 < ktl. This gives our result. 0 LEMMA 3.4. Let a be a nnite measure on R n , let bt = fmn ei <(J,t>da(8),t ERn, and let {8d, i E Z be tbe atoms of a. Tben PROOF. Since IIbt ll 2 = btbt = = ( llln xJin Ln ei<o,t> da(8) In ei<>.,t>da(>.) = ei<O,t>-i<>',t> da(8) x da(>.), 69 we have Now _ 1_ f e i <9->..,t> dt is bounded ' so we may use the Lebesgue Dominant m(Br) Br Convergence Theorem to conclude that o Next we introduce some standard facts for the unitary representation for ]Rn. For detailed treatment, see [Su]. PROPOSITION 3.5. (1) (Stone) If U is a unitary representation of]Rn on a separable Hilbert space, tben tbere exists a spectral measure E in H on tbe set of all Borel sets in ]R n sucb tbat Ut = [ ei <t,9> dE( 8) for all t E ]R n. Jm n (2) (Bochner) If <p is a positive definite function on lR. n, tben tbere exists a unique positive Radon measure a witb total measure a(lRn) = <p(D) sucb tbat o We remark that the above two theorems are equivalent in the sense that one theorem can be easily proved, assuming that the other is true. We also remark that if we fix f E H, then (Ud, f) is a positive definite function on lR n for a unutary representation U of lR. n on H. We let (Ud, f) = fmn ei <t ,9> dO' f( 8). 70 LEMMA 3.6. f E L2(M,m) is an eigenfunction for {Ud corresponding to the eigenvalue ei <80 ,t> iff u / is concentrated at the point 00 ; i.e., PROOF. Since so from the uniqueness part of Proposition 3.5.(2) we obtain Conversely, so I(Ud,!)1 = IIfll2 = IIUdlillfll, so the Cauchy inequality becomes equality. Therefore, Ut!=cd for some constant Ct. But so Ct = e i <8 0 ,t>. LEMMA 0 3 .7. Supose that the JRn Anosov action has a continuous spectrum. Let f E L2(M) , wit}l J fdm = o. Then U/ has no atom. PROOF. Otherwise, u/(Oo) > 0 for some 00 E JRn. But 8(0 - 00 ) is absolutely continuous with respect to u /; we know that there exists a function h E L2 (JR n, U / ) 71 such that o(B - Bo) = h(B)af(Bo). Recall a result in [Su], p.141, that we can find a 9 E< I > (a subspace in L2(M) generated by f) such that a g = 8(B - Bo) (see for example [C-F-S], Appendix 2 for the csae of Anosov diffeomorphisms and Anosov flows). Lemma 3.6 implies that 9 is a non-constant eigenfunction. Contradiction. 0 We are now able to prove the equivalence of (1) and (3) of Theorem 3.2. PROPOSITION 3.8. lim An lR n action bas a continuous spectrum iff r-+oo m (~( r » JrB(r) I(U t I,g)-(j,l)(g,l)ldt=O. PROO F. "=}": A standard polarization trick makes it sufficient to prove lim r-+oo m (~) r I(UtI,!) - (j,l)(l,!)ldt = O. r JBr Moreover, we may assume that (1,1) = 0 by replacing I with I - (1,1). So it is sufficient to show (~) r I(U t I, f)ldt = o. r-+oo m r JBr lim By Lemma 3.3, we know that it is sufficient to prove that lim r-+oo m (~)r JBr r I(U t j, !)1 2 dt = O. But applying Lemma 3.4 to af, and noticing that the above quantity is exactly bt in Lemma 3.4, we conclude our result. The other direction is trivial. 0 We remark that we have actually proved that the equivalence of (1) and (3) is true for general lR n actions, not necessarily Anosov. The proof here is essentially 72 the same as the proof of the same results for diffeomorphisms and for flows , except more complicated technicality. Now we will prove the equivalence of (2) and (1) or (3). First of all we will prove that an irreducible Anosov action cannot have non-constant, measurable eigenfunctions. Recall that JRn Anosov action on M is ergodic [P-S1]. So by [PS2], almost all t E JR n, t is an ergodic diffeomorphism. Take tI, ... , tn E M ergodic, regular with the same foliations as f has and spans JRn, so JRn = ffiJRti. Suppose that H is an eigenfunction for induced, unitary representation. Using a well- known result of Rohlin, we may assume that the following equality is true for all ( tI , ... ,tn ) E JRn and for almost every x E M. U( tl , ... ,tn )H = ei <t1 ,... ,t n l>'1 ,.. . ,>'n> H for some AI, .. . ,An E JR.. LEMMA 3.9. PROOF. For a.e. x E M, H is constant on W:u a.e. The proof is a standard one. There are two approaches to this kind of statement. One is originally used in [AJ. We will use the other approach. Since U t1 H = eit1 >'1 H, without loss of generality we assume that A =1= O. Then U( 27rt l/lt) = H, so H is invariant under the regular element 9 = 27rtI/1l. We define 1nv(g) to be the set of all integrable, g-invariant functions M .-.. JR. We are trying to prove that every function in it is constant almost everywhere on almost every unstable leaf. We define a projection 19 : LI(M).-.. 1nv(g) by then the limit exists almost everywhere and is integrable, and eP .-.. 19eP is a continuous linear map onto the In v(g ). Moreover, the limit 1 r;eP(x) = n I I ~ eP(gkx) n-+±oo n + 1 ~ lim k=O 73 exists almost everywhere and I;</>(x) = Ig</>(x) for almost all x. That is, I; I; = 19 as maps Ll(M) _ 1nv(g). Since the continuous functions are dense in Ll(M), their 19-images are dense in 1nv(g). We now prove that for the continuous function </>, 19</> is essentially constant along FUu, Faa. For any x, y E W;u and any continous </> : M - JR, it is clear that either both 1;(x),I;(y) are defined, or neither, and if defined they are equal. Since T;-</> is defined almost everywhere, I; </> is defined and constant on almost all FUU leaves. Since FUU is absolutely continuous [P-Sl] and I;-</> = I g</> almost everywhere, I g</> is essentially constant on almost every FUU leaf. Similarly, for Fas. The density of the image of continuous functions implies that our function H is essentially constant along FUu, Fas. 0 LEMMA 3.10. H is continuous on a.e. W:, W:. PROOF. From U( tl, ... ,tn )H = e i <tl, ... ,t n l>'I, ... ,>'n> H , it is clear. 0 LEMMA 3.11. H is a constant function almost everywhere. PROOF. We claim that H almost everywhere coincides with some continuous function. The approach is exactly the same as [A] p169. (We do not copy the argument from there; instead, we sketch the idea below. Since H "is" continuous on "every" FU leaf and constant on "every" F SS leaf, hence H "is" continuous.) Now let H' be that continuous function; then it is clear that H' is constant along the leaves of foliation FUu. Hence, H' is constant. 0 Combining the above Lemmas, we proved that (2) implies (1) and (3). Next, we will prove that the other direction is also true. LEMMA 3.12. If Anosov JRn action has a continuous spectrum, then the action is irreducible. 74 PROOF. Otherwise, let p be a periodic point, J{ = W;u i= M. Fix a 1- parameter subgroup gt E lR n that does not fix J{ for some t , and gtoJ( = K. Choose n -1 other elements t1, ... ,t n - 1 in lR n such that p is a common fixed point for them, and together with t they span lRn. It is clear that if we denote by lR n- 1 the space spanned by t 1 , ... , tn-I, then J{' = lR n-1 J{ is compact and is a proper subset of M. Let a function H be defined such that HIKf = 1 HlgtKf = e 2 rr:i(t/to). Then H is a non-constant eigenfunction for Ut . Contradiction. 0 75 4. Metric Transitivity In this section, we will prove the metric transitivity of Anosov lR n actions, or roughly speaking, for any regular element, the strong stable and strong unstable foliations are measurably indecomposable. There are several consequences of this result. The interesting one is that we obtain that ergodicity of regular individual elements. Different from the result of [P-Sl]' where every element off count ably many hyperplanes is ergodic for Anosov actions, we can only have the ergodicity for regular elements. It seems to be a difficult problem to prove that the regular elements are dense even in lR n. We remark that the problem raised by A. Katok fits well into the investigation of rigidity of lR n actions and is one of the motivations for me to study the dynamics of lR n actions. Our method follows closely that of Anosov. Some results of Anosov can be used directly without any change, while others may be proved using his idea. The only difference is that we fit the lR n situation well into the original argument. First we will give the precise definition of metric transitivity. DEFINITION 4.1. A foliation F is called metrically transitive if for an arbitrary measurable F-saturated set A c M, either m(A) = 0 or m(}.1 - A) = O. The idea of the proof of metric transitivity is simple. 0 vVe assume that we have an intermediate saturated set A and prove that the union of all short orbits starting from A is still an intermediate saturated set. Then we use a result that when you iterate A "backwards" sufficiently far away, "all" stable leaves will be contained in the iterated set. Hence, the iterated set has to "have" full measure. But our action is a measure-preserving action; hence A itself "has" full measure. This presents a contradiction. 76 LEMMA 4 .2. Let A be an FUU-saturated Borel set. Let Then it is measurable, constant on the leaves of FUU and lim [ Imr(w) -lldw = o. r-O}A PROOF. Measurability follows from the Fubini theorem, since the set {t : It I ~ r, <ptw E A} is an intersection of the pre-image of the Borel set A under the smooth map M x B(r) - t lvI, (w, t) 1-+ <I>tW, with {w} x B( r). Since A is FUU-saturated and since <I>t permutes the leaves in A, it is clear that m r ( w) is constant on the leaves of FUu. It is clear that where XA is the characteristic function of the set A. We have L m(~(T)) L(l(T) XA(91,W)XA(W)dt) dw = m(~(r)) l(r) (1M XA(<I>tW)xA(W)dW) dt. mT(w) = We show that limt_o iM XA(<PtW)xA(W)dw = meA). Indeed, for any function 1 E L2(M), we have limt_o lI<I>d - 111£2 = 0; hence, Hence, lim r-O 1 A mr(W) = meA). 77 But obviously 0 :s; m r ( w) :s; 1, so our result follows. 0 We remark that we may choose a smooth metric on M, such that the IR n action on an individual orbit is an isometric action. Indeed, pick n 1-dimensional parameter subgroups of IRn that generate IRn; these subgroups give n vector fields on M. These vector fields are linearly independent at every point on M. We define a metric on M such that these vector fields are pairwisely orthogonal and have norm 1. It is then clear that IR n action on the individual orbit is isometric action. LEMMA 4.3. Let w, w' E M such that w' = <Psw, s :s; T. Define . m(Tr s) hm ' r ...... O m(Br) > c> 0 , which is independent of w, w' . PROOF. Using remark above we may assume that we are working on space M = lR n with a flat metric, which is the orbit (up to covering) containing W,W'. The action is the usual translation. Then everything is simple. We draw two T-balls around wand w', the intersection is Tr,s, which contains Tr,r' vVhat is left is an exercise for calculus. LEMMA 4.4. If A is an FUU-saturated, measurable set of M with an interme- diate measure, i.e., 0 < meA) < m(M), then there exists an FUlL-saturated Borel set B C A with an intermediate measure such that 78 is also an FUU-saturated, measurable set with an intermediate measure for some small T. PROOF. First of all we may assume that A itself is a Borel set. In other words, A contains a Borel subset of the same measure, such that the subset is also FUu_ saturated. This fact is proved in [A]. Let D = M - A. We define m r ( w) as in the last lemma and define nrC w) as 1 nr(w) = m(B(r))m{t: It I :::; r,clltw ED}. Lemma 4.2 implies that for sufficiently small r > 0, and c as in the last lemma, the sets B = {w : w E A,mr(w) > 1- c/10}, E = {w: wE D,nr(w) > 1- c/10} have a positive measure and are FUU-saturated. We show that Ene = <p. Otherwise, there would exist a point w E M and s E 1R n such that lsi:::; r, wEB, w' = clls E E. But this is impossible. Indeed, let us calculate the measure of Tr,s as defined in the last lemma. Lemma 4.2 tells us that . c m({tE Tr,s,clltw ED}):::; m({t E B(r),clltw ED}):::; 10m(B(r)). For the same reason we get c m({t E Tr,s,clltw' E A}):::; m({t E B(r) , clltw' E A}):::; 10m(B(r)). But AU D = M, so for every t E Tr,s, clltw = cllt-scllsw' is either in A or in D. So the total measure of Tr,s is less than 2 1CO m(B(r)) :::; kcm(B(r)), and contradicts Lemma 4.3. So C n E = <p. 79 After possibly removing from B some set of m easure zero, we may assume that B is a Borel set. Then C is measurable because it is the image of the Borel set B X B( T) under a smooth map. It is clear that C is FUU-saturated, and has an D intermediate measure. We quote a modified result of Anosov [AJ. LEMMA 4.5. Fix a regular element j E ]Rnj let gt be the l-parameter subgroup containing j with gto = j, to > O. Let A be an arbitrary, measurable set. For any r > 0, c > 0 and sufficiently large t > 0, there exists a set M; such that mOWn < c and any two points w, Wi E M - M:, which lie on the same leaf of the strong stable foliation F88 at distance < r from each other; either both belong or both do not belong to the set g-tA. We remark that the proof in [AJ uses only the exponential contraction property of the foliation F88. So all the proof goes through without any modification. Moreover, the proof is clearly true if we replace the I-parameter subgroup gt by a small cone centering around gt. -VVe require only that the cone be small enough to have a uniform estimate for the exponential contraction. We need one more technique lemma. LEMMA 4 .6. Let]R n Anosov action be irreducible. Fix measurable sets B, U1 , ... , UN of positive measure and also fix a small cone around gt as in Lemma 4.5. We call the half-cone containing j the positive half-cone, and denote it by T+. Then there exists a sequence tj = (ti ), ... , t~)) E T+ with Itil i 00 and a 8 > 0 such that for all tn PROOF. From Proposition 3.8 it is easy to see that for each Uj there exists a 80 Ji C Rn of zero density such that lim _ oo m(<Pt, n Ui) = m(B)m(Ui). We take 1tl Uf.J; J = It u· .. u J N; then this set has zero density, and our result follows easily. THEOREM 4.7 (METRIC TRANSITIVITY). 0 Weakly mixing Anosov Rn action is metrically transitive. PROOF. We wish to arrive at a contradiction on assuming the existence of the sets B and C and the number T of the Lemma 4.4. It is clear that under this assumption there exists an r > 0 such that if WEB, then the r-neighborhood of the point w on the leaf of the foliation FU passing through this point is entirely contained in C. This r is dependent on T, but not on B; hence for the sets <PtB and <PtC, which obviously have the same properties as Band C, we can use this same number r (since T is the same for these). We consider a finite atlas {(Uj, <Pi)}, i = 1,2, ... ,N, of the manifold M. Each coordinate neighborhood Uj = <P i 1 (X x Y) is sufficiently small such that an rneighborhood of any point W E Uj on the leaf of the foliation FU passing through it contains the connected component N~ of the intersection of this leaf with Uj. Note that if W has coordinates <Pi( w) = (x, y), then <p(N~) = x x Y. The <Pi is absolutely continuous. The existence of these kinds of product neighborhoods is proved in [A] for any two foliations that have smooth individual leaves and that are absolutely continuous, transversal to each other. We apply Lemma 4.7 to the sets B, U1 , .•. , UN. This lemma and the absolute continuity of <Pi guarantee the existence of a sequence in --+ 00, i E T+ and a number ~ > 0 such that m( <Pi( <P- tn B n Ui)) > 8, i = 1, ... , N. This and the properties of the set C imply that for some 8 > 0 We now use Lemma 4.5 and the remark after it, replacing A by C and assuming 81 that the number r in this lemma exceeds the size of leaves of the foliation FSS n Ui. For sufficiently large Itn I the measure of the "exceptional" set Nf!n of the Lemma 4.5 will be arbitrarily small. That means that the measure of the set <Pi(M!n n Ui) can also be considered arbitrarily small, significantly smaller than 8. Hence, for each of these sets, if Itn I is sufficiently large, there exists an Xn such that and m(<pi(M;n n Ui) n (xn X Y» is small. Now if the point (x,y) lies outside the set then it belongs or does not belong to <PiC <P -tn n Ui) simultaneously with the points (xn,y), and the last one belongs to <Pi(<I>-t n n Ui). Therefore, and hence also m(Ui - <I>-t nC) ---+ 0 as n ---+ 00. Therefore, m(M - C) = m(M - <I>-t n C) ~ L m(U i - <I>-t nC) ---+ O. I But this contradicts the fact that the measure of C is intermediate. COROLLARY 4.8. 0 Let tbe IRn action be irreducible. Tben (1) every regular elements are ergodic; (2) every I-parameter regular subgroup gt is weakly mixing. PROOF. (1) Let Uf = F or f(g:r) = f(x) for a regular element and a.e x E .1\1. Consider h = Re(f); it is clear that h(g.c) = hex) a.e. Then Wo := {x : f( x) ~ a} is <P invariant. But it is easy to see that h is constant on a.e. Fuu-Ieaf, so WCJ is FUU-saturated. SO WCJ = AI or 9. This forces h to be essentially constant. 82 The same argument applies to the imaginary part of f. Finally, we get f as essentially constant. (2) f(gtx) = ei>..tf(x) implies that f(g27r/>"X) = f(x), but g27r/>" is regular. Our result follows from (1). 0 83 5. Measures Compatible with R n Actions There are two natural measures compatible with Anosov diffeomorphisms and Anosov flows, which are called the Sinai-Bowen-Ruelle measure and the BowenMargulis measure. These two measures are considered extensively by various authors, and it is clear that they play an important role in the rigidity investigation. They will be proved to exist for Anosov R n actions as well. THEOREM (THE EXISTENCE OF THE SBR MEASURE) 5.1. Let the Rn action be irreducible, hence weakly mixing. Fix a regular element and then the corre- sponding strong unstable foliation FUu. Then there exists an invariant measure p such that the conditional measure with respect to FUU is absolutely continuous with respect to the Lebesgue measure. PROOF. Fix a regular element 9 E Rnj then 9 expands the foliation F = FUu, the leaves of which are smooth. Fix a positive integer r. Choose a covering of M by a finite number of charts D m - k x Dk such that the leaves of F are of the form Dm-k x {v}. Let a probability measure p on M have, to each of these charts, a restriction of the form where du is the Lebesgue measure on Dn-k and p( dv) some positive measure on Dk. Assume that for p-almost all v, the function u --+ poe u, v) is strictly positive, and its logarithm has derivatives up to r which are Lipschitz, with Lipschitz constant:::; 1. Let K be the set of such measures p, with 1 fixed, but p( dv) are allowed to vary. Define K+ = Ul~O nn~O g1t K(l). It is easy to verify that K+ does not depend on the choice of the charts used to define K(l). 84 Then we use the theorem in [Rl], obtaining the following: (a) K is vaguely compact; it is a Choquet simplex, and the conditional measure pO(u,v)du of p E K+ on a leaf of:F is independent of p (up to normalization). (b) K+ is non-empty. (c) Let Kg be the set of g-invariant elements of K+; then Kg is a simplex. It is also straightforward that the following statement is true. Claim: (d) if p. E K+, then I p. E K for any <p E 1R n. (e) K+ is convex. Indeed, in a coordinate chart, let 1-1 = (h(u,v),h(v)); then for some a ~ O. So the density of the conditional measure corresponding to I p. on D m - k x {v} is 0"1 = K O"°(fl (u, v), h( v ))1 det(Duh (u, v))I, and so Du 10gO"I(u, v) = Du log O"°(h (u, v), h( v ))Duh (u, v) + Du log 1det(D h (u, v ))1 ~ ~ aDulogO"°(h(u,v),h(v)) + C. So for p.-almost all v, Same calculations (as those in [Rl]) conclude that there exists [' such that hl E Kif. SO Ip. E nlgnK(l) = ngn(fK(l)) c ngn(K(l')). SO Ip. E K+. This proves (d). (e) is clear. 85 Because (a)-(e), we may use first use an average method in K+ to get that Kg is non-empty, and then to use the average method to get that in there exists an JRn-invariant measure in Kg. (e) and (a) guarantee that they are in the right class of measures. 0 Margulis considered an invariant measure for anosov flows whose conditional measure restricted to FUU ,FsS have uniform expansion (contraction) property. Moreover, the coefficients are both topological entropy for the flow. There are also several other constructions, e.g., Sinai lSi] for toral diffeomorphisms, Hasselblatt [Ha] for Anosov flows, Hamenstiidt [Ham] for geodesic flows for negatively curved manifolds, which are all interesting. In this paper we will give a similar fact about this kind of measure, and we will call it Margulis measure. DEFINITION 5.2. Fix a regular element f E jRn.We call an invariant measure p a Margulis measure if (1) g*puu = )..(g)pUU, for all 9 E jRn, some )..(g). (2) pUu is honolomy-invariant. The same formulas hold for pSs. THEOREM (THE EXISTENCE OF THE MARGULIS MEASURE) 5.3. For any irre- ducible Anosov jR n action, there exists at least one Margulis measure. PROOF. A similar argument in [R2] establishes the existence of transversal holonomy-invariant measure with uniform expansion in the Fuu-direction. For completeness, we will reproduce the proof with necessary modification. Let F = FUu, with codim(F) = k. Let S denote the set of open submanifolds of dimension k transversal to F. Let .J = {p : p is a signed transversal-invariant measure for fl. 86 (A transversal-invariant measure is a collection (PE) of real measures on the set I: E S such that under the canonical isomorphisms they coincide.) We define vague topology as the topology defined on :1 by the seminorms: where ¢> is a real, continuous function with compact support in I: E S. Call P :?: 0 if PE :?: 0 for all I: E S. Let C be the cone of positive measures in:1. Then we have (see [R2]) (a) g:1 = .1, gC = Cj (b) there exists a .\0 :?: 0, such that C (.\0, g) {p E C gp - .\op} IS a non-empty, closed subset of C. Let us take a 1-parameter subgroup gt in lR n with gl = 9 and show that the set C(.\o, {gt}) = {p E C : gp = .\~p for all t} is non-empty and g-invariant. Indeed, C(.\o, gl/2) C C(.\o,g), and g-invariance is obvious. Then repeating the argument again and again, we may prove that C(.\o, gl/2n+1) C C(,\o, gl/2n) and is also g-invariant. Local compactness of C implies that the intersection Co of these sets is non-empty and is g-invariant. Now the continuity of gtll in t with Il E Co implies our result. So we have that (c) C(.\o, {gd) = {p E C : gp = .\~p for all t} is a non-empty and g-invariant set. It is clear that Co is a closed sub cone of C, and for all f E ]R n fC o = fC o. So using the argument of [R2], we can get that C(.\l' J) = {p E Co : f p = .\lp} is a non-empty, closed subset of Co for f sufficiently close to 9 such that f is regular and expands F. Then we may have a statement similar to (c). Continuing to do this, we will obtain a measure that is a transversal holonomy-invariant measure with uniform expansion property for all elements for ]Rn. Using Margulis' 87 construction of Margulis measure for Anosov flow, we will get a Margulis measure for our Anosov IR n action. 0 Remark 5.4. The argument for the two above theorems is almost the duplication of those in [Rl] [R2]; we do not have uniqueness automatically from the argument of Ruelle [Rl] [R2]. Yet, we do have the uniqueness of the uniform expansion coefficient for Margulis measure. This fact can be proved using either Margulis' original argument [M2] or Hasselblatt's construction for the Margulis measure [Ha], which we do not discuss here. We also remark that the uniqueness of the Margulis measure might be important for the proof of the rigidity of IR n actions, because the Margulis measure is carried over by topological conjugacy. 88 References [A] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, (English translation, A.M.S.,Providence,RI, 1969), Proceedings of the Steklov Institute of Mathematics 90 (1967), 1-235. [A-S] L. Auslander and J. Scheuneman, On certain automorphisms of nilpotent Lie groups, in A.M.S. 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