FAILURE OF THE POINTWISE AND MAXIMAL ERGODIC THEOREMS FOR THE FREE GROUP

Let $F_{2}$ denote the free group on two generators $a$ and $b$. For any measure-preserving system $(X,{\mathcal{X}},{\it\mu},(T_{g})_{g\in F_{2}})$ on a finite measure space $X=(X,{\mathcal{X}},{\it\mu})$, any $f\in L^{1}(X)$, and any $n\geqslant 1$, define the averaging operators $$\begin{eqnarray}\displaystyle {\mathcal{A}}_{n}f(x):=\frac{1}{4\times 3^{n-1}}\mathop{\sum }_{g\in F_{2}:|g|=n}f(T_{g}^{-1}x), & & \displaystyle \nonumber\end{eqnarray}$$ 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.