University of Illinois Urbana-Champaign

Pointwise ergodic theorems via descriptive set theory

Zomback, Jenna

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ZOMBACK-DISSERTATION-2022.pdf

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https://hdl.handle.net/2142/115541

Description

Title
Pointwise ergodic theorems via descriptive set theory
Author(s)
Zomback, Jenna
Issue Date
2022-04-13
Director of Research (if dissertation) or Advisor (if thesis)
Tserunyan, Anush
Doctoral Committee Chair(s)
Boca, Florin
Committee Member(s)
  • Junge, Marius
  • Hieronymi, Philipp
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • descriptive set theory
  • ergodic theory
Abstract
We prove several new instances of the pointwise ergodic theorem through the lens of descriptive set theory. Additionally, we provide short, elementary proofs of certain existing pointwise ergodic theorems. These results equate the global condition of ergodicity with the local (pointwise) statistics of the action. This interplay lends itself towards finitary and self-contained descriptive set theoretic arguments, whereas many classical proofs of pointwise ergodic theorems use operator ergodic theorems as black boxes. All of our proofs will center around the fact that we may replace the integral of a function with the integral of the function averaged over a finite set. This ``local-global bridge" distills out the analytic part from the proofs of pointwise ergodic theorems, reducing them to combinatorial (finitary) tiling problems. In Chapter 2, in joint work with Jon Boretsky, we solve the aforementioned tiling problem for probability measure preserving (pmp) actions of amenable groups along increasing Tempelman F{\o}lner sequences, thus providing a short and combinatorial proof of the (already well known) corresponding pointwise ergodic theorem due to Tempelman. Chapters 3 and 4 are based on joint work with Anush Tserunyan. In Chapter 3, we prove a new backward ergodic theorem for a countable-to-one pmp transformation $T$, where the averages are taken along trees in the backward orbit of the point (i.e.~along trees of possible pasts). As a corollary, we prove a new pointwise ergodic theorem for pmp actions of free groups, where the ergodic averages are taken along arbitrary subtrees of the standard Cayley graph rooted at the identity, strengthening an earlier (from 2000) theorem of Bufetov in the case of group actions. We also discuss other applications of this backward theorem, in particular to the shift map on Markov measures, which yields a pointwise ergodic theorem along trees for the action of the free groups on their boundary. The backward ergodic theorem for a transformation $T$ can be regarded as a pointwise ergodic theorem for certain kinds of partial injective transformations (Borel right inverses of $T$). In Chapter 4, we generalize this theorem to apply to a wider class of quasi pmp actions of countable free semigroups, without the injectivity assumption.
Graduation Semester
2022-05
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/115541
Copyright and License Information
Copyright 2022 Jenna Zomback

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