Pointwise Ergodic Theory: Examples and Entropy

The Oseledets Multiplicative Ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.

We prove Bourgain's Return Times Theorem for a range of exponents $p$ and $q$ that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain's averages. As an immediate corollary we obtain a Wiener-Wintner type of result for the ergodic Hilbert series.

Ergodic optimization is the study of extremal values of asymptotic dynamical quantities such as Birkhoff averages or Lyapunov exponents, and of the orbits or invariant measures that attain them. We discuss some results and problems.

In this work we prove the pointwise ergodic theorem for harmonic degree 1 cocycle of a measurable stationary action of Z^d on a probability space. In a precedent paper Boivin and Derriennic (1991) studied this theorem for not necessarily harmonic cocycles. The harmonic hypothesis allows, in the elliptic case, to change the integrability condition to L^2, while Boivin and Derriennic showed that the optimum condition in the non-harmonic case is the finiteness of Lorentz's norm ...

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $\rho$-variational inequalities on $L^2$ for $2<\rho<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodi...

In this diploma thesis (written in German) we present a detailed proof of Bourgain's Return Times Theorem due to Bourgain, Furstenberg, Katznelson and Ornstein following their paper as well as the book by Assani. Moreover, we generalize the result to return time sequences coming from systems with purely atomic invariant $\sigma$-algebra.

In this paper, we study the pointwise convergence of some continuous-time polynomial ergodic averages. Our method is based on the topological models of measurable flows. One of main results of the paper is as follow. Let $(X,\mathcal{X},\mu, (T^{t})_{t\in \mathbb{R}})$ and $(X,\mathcal{X},\mu, (S^{t})_{t\in \mathbb{R}})$ be two measurable flows, $a\in \mathbb{Q}$, and $Q\in \mathbb{R}[t]$ with $\text{deg}\ Q\ge 2$. Then for any $f_1, f_2, g\in L^{\infty}(\mu)$, the limit \beg...

Ergodic Functions are bounded uniformly continuous $(\text{BUC})$ functions that are typical realizations of continuous stationary ergodic process. A natural question is whether such functions are always the sum of an almost periodic with an $L^1-$mean zero $\text{BUC}$ function. The paper answers this question presenting a framework that can provide infinitely many ergodic functions that are not almost periodic plus $L^1-$ mean zero.

It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb{Z}^d$ on a probability space $X$ and a nonnegative measurable function $f$ on $X$, the probability that the sequence of ergodic averages $$ \frac 1 {(2k+1)^d} \sum\limits_{g \in [-k,\dots,k]^d} f(g \cdot x) $$ has at least $n$ fluctuations across an interval $(\alpha,\beta)$ can be bounded from above by $c_1 c_2^n$ for some universal constants $c_1 \in \mathbb{R}$ and $c_2 \in (0,1)$, ...

This short note reviews the basic theory for quantifying both the asymptotic and preasymptotic convergence of Markov chain Monte Carlo estimators.