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

Lewis Bowen; Peter Burton

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

Dynamical Systems 2019-08-02 v3

Authors: Lewis Bowen , Peter Burton

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 $p=1$ .

Keywords

pointwise ergodic theorem PSL_2(R) ball averages measure-preserving action L^1 failure

Cite

@article{arxiv.1901.01299,
  title  = {Failure of the $L^1$ pointwise ergodic theorem for $\mathrm{PSL}_2(\mathbb{R})$},
  author = {Lewis Bowen and Peter Burton},
  journal= {arXiv preprint arXiv:1901.01299},
  year   = {2019}
}

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