Topics in descriptive set theory related to equivalence relations, complex borel and analytic sets
<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 Geome...
| Summary: | <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<sub>∞</sub>-
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 ≤ α < ∞, the set of entire
functions whose order is equal to α is ∏<sup>0</sup><sub>3</sub>-complete and the set of all sequences
of entire functions whose orders converge to α is ∏<sup>0</sup><sub>5</sub>-complete. We also prove
that given any line in the plane and any cardinal number 1 ≤ n ≤ N<sub>0</sub>, the
set of continuous paths in the plane tracing curves which admit at least n
tangents parallel to the given line is Σ<sup>1</sup><sub>1</sub>-complete and the set of differentiable
paths of class C<sup>2</sup> in the plane admitting a canonical parameter in [0,1] and
tracing curves which have at least n vertices is also Σ<sup>1</sup><sub>1</sub>-complete.</p> |
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