Failure of the $L^1$ pointwise ergodic theorem for $\mathrm{PSL}_2(\mathbb{R})$
arXiv:1901.01299
summary
The authors construct an explicit example showing that the pointwise ergodic theorem for measure‑preserving actions of PSL₂(ℝ) fails in the L¹ setting, contrary to the known results for 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 $p=1$.
4 figures added
Topics & keywords
#ergodic theory#pointwise ergodic theorem#PSL(2,R) actions#L¹ counterexample#measure-preserving dynamicspointwise ergodic theoremL¹PSL(2,R)ball averagesmeasure-preserving action