Structures and dynamics

"Our results are divided in three independent chapters. In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: ""For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M)."" From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15]. In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16]. In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16]."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Panagiotopoulos, Aristotelis

Contributors dc:contributor

• Solecki, Slawek
• Hieronymi, Philipp
• Tserunyan, Anush
• van den Dries, Lou

• Polish groups
• Fraisse
• Turbulence
• Hjorth
• Left invariant
• Becker
• Projective Fraisse
• Infinite games
• Borel complexity

Statement dc:rights

• Copyright 2017 Aristotelis Panagiotopoulos

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/98378

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/98378