Topics in descriptive set theory related to equivalence relations, complex borel and analytic sets

Author: Sofronidis, Nikolaos Efstathiou

Year: 1999

Degree: Dissertation (Ph.D.)

Advisors: Kechris, Alexander S.; Wolff, Thomas H.

Committee Members: Kechris, Alexander S.; Luxemburg, W. A. J.; Wolff, Thomas H.; Gao, S.

Option: Mathematics

DOI: 10.7907/vva8-1959

Abstract

The purpose of this doctoral dissertation is first to show that certain kinds of invariants for measures, self-adjoint and unitary operators are as far from complete as possible and second to give new natural examples of complex Borel and analytic sets originating from Analysis and Geometry.

The dissertation is divided in two parts.

In the first part we prove that the measure equivalence relation and certain of its most characteristic subequivalence relations are generically S∞- ergodic and unitary conjugacy of self-adjoint and unitary operators is generically turbulent.

In the second part we prove that for any 0 ≤ α < ∞, the set of entire functions whose order is equal to α is ∏03-complete and the set of all sequences of entire functions whose orders converge to α is ∏05-complete. We also prove that given any line in the plane and any cardinal number 1 ≤ n ≤ N0, the set of continuous paths in the plane tracing curves which admit at least n tangents parallel to the given line is Σ11-complete and the set of differentiable paths of class C2 in the plane admitting a canonical parameter in [0,1] and tracing curves which have at least n vertices is also Σ11-complete.

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