Electronic Theses and Dissertations

Date

2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Máté Wierdl

Committee Member

Béla Bollobás

Committee Member

James T. Campbell

Committee Member

Randall McCutcheon

Committee Member

Máté Wierdl

Abstract

\hspace{.2cm} In the first part of our thesis, we introduce the `grid method' to prove that a very strong type of oscillation occurs for the averages obtained by sampling a flow along the sequence of times of the form $\{n^\alpha: n\in \setN\}$, where $\alpha$ is a {positive non-integer rational number}. More precisely, we show that in every aperiodic dynamical system $(X, \Sigma, \mu, T^t)$, we can find a set $E$ of arbitrarily small measure so that $\limsup_N\frac1N \sum_{n\le N}\setone_E(T^{n^\alpha}x)=1$ and $\liminf_N\frac1N \sum_{n\le N}\setone_E(T^{n^\alpha}x)=0$ almost everywhere. Such behavior of a sequence is known as the \emph{strong sweeping out} property. Using the grid method, we will give an example of a general class of sequences which satisfy the {strong sweeping out} property.\\ \hspace{.2cm} In the second part of our thesis, we deal with ergodic averages along a \emph{sublacunary sequence} of positive integers. An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+\eta$ for some positive $\eta$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out. As a further application of the grid method, we improve this result by showing that if $(a_n)$ satisfies only $\frac{a_{n+1}}{a_n}>1+\frac1{(\log\log n)^{1-\eta}}$ for some positive $\eta$ then it is already strong sweeping out.\\ \hspace{.2cm} In the third part, we will prove that if an integer sequence $(b_n)$ grows slower than any positive power of $n$, and $(a_n)$ is a small perturbation of $(b_n)$, then $(a_n)$ is strong sweeping out. This result is sharp. That means for any $c>0$, the ergodic averages along $\lfloor n^c \rfloor$ converge almost everywhere for all $f\in L^p$ when $p>1$.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest

Notes

Open Access

Download

Share

COinS