Failure of the $L^1$ pointwise and maximal ergodic theorems for the free group

TL;DR

This paper demonstrates that the pointwise and maximal ergodic theorems fail in $L^1$ for free group actions, providing a counterexample that contrasts with positive results in higher $L^p$ spaces.

Contribution

It constructs a specific measure-preserving system and function showing unbounded ergodic averages in $L^1$, highlighting limitations of ergodic theorems for free groups.

Findings

Counterexample in $L^1$ for free group actions

Failure of pointwise and maximal ergodic theorems in $L^1$

Contrast with positive results in $L^p$, $p>1$

Abstract

Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},\mu)$, any $f \in L^1(X)$, and any $n \geq 1$, define the averaging operators $${\mathcal A}_n f(x) := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x ),$$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f \in L^1(X)$ such that the sequence ${\mathcal A}_n f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^1$ for actions of $F_2$. This is despite the results of Nevo-Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^p$ for $p>1$ and for $L \log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell^1(F_2)$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.