FAILURE OF THE POINTWISE AND MAXIMAL ERGODIC THEOREMS FOR THE FREE GROUP
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 $|g|$ …