Dynamics of Non-Classical Interval Exchanges

Author: Gadre, Vaibhav S.

Year: 2010

Degree: Dissertation (Ph.D.)

Advisors: Calegari, Danny C.; Dunfield, Nathan M.

Committee Members: Calegari, Danny C.; Dunfield, Nathan M.; Ramakrishnan, Dinakar; Ni, Yi

Option: Mathematics

Abstract

Train tracks with a single vertex are a generalization of interval exchange maps. Here, we consider non-classical interval exchanges: complete train tracks with a single vertex. These can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff's simplicial systems. We show that the refinement process gives an expansion that has a key dynamical property called uniform distortion. We use uniform distortion to prove normality of the expansion. Consequently, we prove an analog of Keane's conjecture: almost every non-classical interval exchange is uniquely ergodic. In the concluding chapter, we state an application of the main results of the thesis to a question about harmonic measures on the Thurston boundary of Teichmuller space coming from finitely supported random walks on the mapping class group.

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