Dissertation
Cohomological properties of Borel automorphisms and substitutions on infinite alphabet
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2021
DOI: 10.17077/etd.006049
Abstract
This thesis is divided into two parts. In the first part we study cohomology of countable groups of Borel automorphisms of a standard Borel space. We prove that for a hyperfinite group of Borel automorphisms, the subgroup of coboundaries is dense in the group of cocycles. We study the cohomology of the 2-odometer from Borel perspective and describe all cocycles of 2-odometer taking values in a locally compact second countable group G. We show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup H of G. We study cocycles of a minimal homeomorphism of Polish space and provide a Borel version of Gottschalk-Hedlund theorem in this context. We study sets of real valued coboundries of two commuting Borel automorphisms. We show that if S,T are two commuting automorpisms of standard Borel space, such that they generate a free Borel Z^2-action then S and T do not have same sets of real valued coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel Z^d-actions.
In the second part we study substitution dynamical systems on infinite alphabets from the perspective of Borel dynamics. We prove two versions of Rokhlin’s lemma for subshifts associated with substitutions on infinite alphabets. We construct stationary (and non-stationary) Bratteli-Vershik models for such substitution dynamical systems. Using the Bratteli-Vershik model we give an explicit formula for a shift-invariant measure (finite and infinite) and provide a criterion for this measure to be ergodic (or uniquely ergodic).
Details
- Title: Subtitle
- Cohomological properties of Borel automorphisms and substitutions on infinite alphabet
- Creators
- Shrey Sanadhya
- Contributors
- Palle Jorgensen (Advisor)Sergii Bezuglyi (Advisor)Ionut Chifan (Committee Member)Raúl Curto (Committee Member)Charles Frohman (Committee Member)Lihe Wang (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2021
- DOI
- 10.17077/etd.006049
- Publisher
- University of Iowa
- Number of pages
- ix, 123 pages
- Copyright
- Copyright 2021 Shrey Sanadhya
- Language
- English
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (pages 119-123).
- Public Abstract (ETD)
The study of dynamical systems aims to understand changes that occur in a physical (or abstract) system over time. Some real-life examples include the stock market fluctuations, the dissolution of ink in water and the weather patterns. This thesis is dedicated to creating abstract mathematical models to understand such dynamical behavior. We begin with two objects: an underlying space and a rule to describe how a point in this space changes over time. We can iterate this rule indefinitely to obtain an orbit of a point. One of the goals of the study of a dynamical system is to understand the behavior of orbits and identify if it is similar (isomorphic) to a previously known dynamical system. This allows us to classify dynamical behaviors.
The aim of the first part of this thesis is to classify dynamical systems known as Borel dynamical system using a parameter called cohomology group. We developed methods to identify certain classes of Borel dynamical systems (known as hyperfinite systems and odometers) using their respective cohomology groups. We also provide a criterion to determined when two Borel dynamical systems do not have same sets of certain parameters (known as coboundaries). Thus this work furthers the classification theory of Borel dynamical systems.
The second part of this thesis is dedicated to the study of a class of dynamical systems known as substitution on infinite alphabet. This class of dynamical systems can be realized as Borel dynamical systems. It is known that every Borel dynamical system can be modeled as a graph theoretic object called Bratteli-Vershik model. So a natural question arises: What can a Bratteli-Vershik model tell us about substitutions on infinite alphabet? In this thesis we answer this question by first providing an algorithm to construct Bratteli-Vershik models for substitutions on infinite alphabet. Then using the Bratteli-Vershik model we find key characteristics of substitutions on infinite alphabet for example, expression for an ergodic invariant measures.
- Academic Unit
- Mathematics
- Record Identifier
- 9984097171302771
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