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

• Main Author: Sofronidis, Nikolaos Efstathiou
• Format: Others
• Language: en
• Published: 1999
• Online Access: https://thesis.library.caltech.edu/10021/1/Sofronidis_NE_1999.pdf; Sofronidis, Nikolaos Efstathiou (1999) Topics in descriptive set theory related to equivalence relations, complex borel and analytic sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/vva8-1959. https://resolver.caltech.edu/CaltechTHESIS:01232017-134252080 <https://resolver.caltech.edu/CaltechTHESIS:01232017-134252080>

<p>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.</p> <p>The dissertation is divided in two parts.</p> <p>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.</p> <p>In the second part we prove that for any 0 &#8804; α &#60; ∞, 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 &#8804; n &#8804; 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.</p>