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

## TL;DR

This paper demonstrates that the pointwise ergodic theorem for measure-preserving actions of PSL(2,R) does not hold in L^1, providing explicit counterexamples to Nevo's theorem which applies for p>1.

## Contribution

The paper presents explicit examples showing the failure of the L^1 pointwise ergodic theorem for PSL(2,R), highlighting the limitations of Nevo's theorem.

## Key findings

- The L^1 pointwise ergodic theorem fails for PSL(2,R).
- Explicit counterexamples are constructed.
- Nevo's theorem cannot be extended to p=1.

## 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 $p=1$.