Slowly decaying averages and fat towers

TL;DR

This paper investigates the integrability of a specific stopping time related to ergodic averages and introduces the concept of fat towers in ergodic systems, showing that the stopping time need not be integrable even for bounded functions.

Contribution

It demonstrates that the stopping time defined by the decay of ergodic averages is not necessarily integrable, and introduces fat towers as a new structural element in ergodic systems.

Findings

The stopping time N(x) can be non-integrable even for bounded functions.

Existence of fat towers in every ergodic system.

Counterexample to a question posed by Martin Barlow.

Abstract

Let $(X,\Sigma,m,\tau)$ be an ergodic system, that is, $(X, \Sigma, m)$ is a probability space and $\tau: X \to X$ is an invertible ergodic $m$-preserving transformation. For a function $f:X\to\mathbb R$, let $A_Nf$ denote the $N$th ergodic average, $A_Nf(x)=\frac{1}{N}\cdot (f(x)+\dots+\tau^ {N-1}f(x))$. Martin Barlow (personal communication) asked the following question, which arose from the work of a student (Zichun Ye) on interface models. Question: If $f(x) \ge 0$ is integrable, and $N(x) = \min \{n: A_kf(x) \le 2 \int f \text{for all} k \ge n\}$, is it the case that $N(x)$ is also integrable? In this note we show that the answer to this Question is no in general, even for bounded functions. In so doing we discover that every ergodic system has a special sort of Kakutani tower which we call a fat tower.