Uniformly rigid homeomorphisms

Yancey, Kelly

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https://hdl.handle.net/2142/45507 Copy

Description

Title

Uniformly rigid homeomorphisms

Author(s)

Yancey, Kelly

Issue Date

2013-08-22T16:42:34Z

Director of Research (if dissertation) or Advisor (if thesis)

Rosenblatt, Joseph

Doctoral Committee Chair(s)

Ruan, Zhong-Jin

Committee Member(s)

• Rosenblatt, Joseph
• Erdogan, M. Burak
• Rapti, Zoi

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• weak mixing
• topological weak mixing
• rigid
• uniformly rigid
• ergodic theory
• topological dynamics
• generic
• typical homeomorphisms

Abstract

In this dissertation we are interested in the study of dynamical systems that display rigidity and weak mixing. We are particularly interested in the topological analogue of rigidity, called uniform rigidity. A map $T$ defined on a topological space $X$ is called \textit{uniformly rigid} if there exists an increasing sequence of natural numbers $\left( n_m\right)$ such that $\left( T^{n_m}\right)$ converges to the identity uniformly on $X$ and is called \textit{weakly mixing} if there exists a sequence $\left(s_m\right)$ of density one such that $\mu(T^{s_m}A\cap B)$ converges to $\mu(A)\mu(B)$ for every $A,B$ of positive $\mu$-measure (the sequence $\left(s_m\right)$ is called a \textit{mixing sequence}). Uniform rigidity and weak mixing are two properties of a dynamical system that are very different, though not exclusive. Rigidity implies that at certain times the image of an interval is close to the interval, while weak mixing implies that at other times the images of intervals are evenly distributed. Observe that the rigidity times for a weakly mixing map have density zero. This dissertation attempts to better understand the interplay between weak mixing and uniform rigidity. The underlying theme of this dissertation has two threads: (1) to determine how the topology of a space affects dynamical properties of maps that are defined there and (2) to characterize the structure of uniform rigidity sequences for weakly mixing maps. The work in this dissertation has involved several projects that were designed to provide a better understanding of these maps and their uniform rigidity sequences, thereby yielding information about the dynamical properties that are compatible with certain spaces and information about the structure of those sequences.

Graduation Semester

2013-08

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http://hdl.handle.net/2142/45507

Copyright and License Information

Copyright 2013 Kelly Yancey

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