Failure of the $L^1$ pointwise ergodic theorem for ${\mathrm{PSL}}_2(\mathbb{R})$

Abstract

Amos Nevo established the pointwise ergodic theorem in $L^p$ for measure-preserving actions of ${\mathrm{PSL}}_2(\mathbb{R})$ on probability spaces with respect to ball averages and every $p>1$. This paper shows by explicit example that Nevo's Theorem cannot be extended to $p=1$.

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