Separating measurable recurrence from strong recurrence via rigidity sequences
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arXiv
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866910762476240896 |
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| author | Griesmer, John T. |
| author_facet | Griesmer, John T. |
| contents | If $G$ is an abelian group, we say $S\subset G$ is a set of recurrence if for every probability measure preserving $G$-system $(X,μ,T)$ and every $D\subset X$ having $μ(D)>0$, there is a $g\in S$ such that $μ(D\cap T^{g}D)>0$. We say $S$ is a set of strong recurrence if for every set $D$ having $μ(D)>0$ there is a $c>0$ such that $μ(D\cap T^{g}D)>c$ for infinitely many $g\in S$. We call $S$ measure expanding if for all $g\in G$, the translate $S+g$ is a set of recurrence. A rigidity sequence for $(X,μ,T)$ is a sequence of elements $s_n\in G$ satisfying $\lim_{n\to\infty} μ(D\triangle T^{s_n}D)=0$ for all measurable $D\subset X$.
For all but countably many countable abelian groups $G$, we prove that if $S$ is measure expanding, there is a sequence of elements $s_n\in S$ such that $\{s_n:n\in \mathbb N\}$ is also measure expanding and every translate of $(s_n)$ is a rigidity sequence for some free weak mixing measure preserving $G$-system. The special case where $S=G$ proves a conjecture of Ackelsberg.
As a consequence, we prove that for every countably infinite abelian group $G$ and every measure expanding set $S\subset G$ there is a subset $S'\subset S$ such that $S'$ is measure expanding and no translate of $S'$ is a set of strong recurrence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_05609 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Separating measurable recurrence from strong recurrence via rigidity sequences Griesmer, John T. Dynamical Systems 11B30, 37A45 If $G$ is an abelian group, we say $S\subset G$ is a set of recurrence if for every probability measure preserving $G$-system $(X,μ,T)$ and every $D\subset X$ having $μ(D)>0$, there is a $g\in S$ such that $μ(D\cap T^{g}D)>0$. We say $S$ is a set of strong recurrence if for every set $D$ having $μ(D)>0$ there is a $c>0$ such that $μ(D\cap T^{g}D)>c$ for infinitely many $g\in S$. We call $S$ measure expanding if for all $g\in G$, the translate $S+g$ is a set of recurrence. A rigidity sequence for $(X,μ,T)$ is a sequence of elements $s_n\in G$ satisfying $\lim_{n\to\infty} μ(D\triangle T^{s_n}D)=0$ for all measurable $D\subset X$. For all but countably many countable abelian groups $G$, we prove that if $S$ is measure expanding, there is a sequence of elements $s_n\in S$ such that $\{s_n:n\in \mathbb N\}$ is also measure expanding and every translate of $(s_n)$ is a rigidity sequence for some free weak mixing measure preserving $G$-system. The special case where $S=G$ proves a conjecture of Ackelsberg. As a consequence, we prove that for every countably infinite abelian group $G$ and every measure expanding set $S\subset G$ there is a subset $S'\subset S$ such that $S'$ is measure expanding and no translate of $S'$ is a set of strong recurrence. |
| title | Separating measurable recurrence from strong recurrence via rigidity sequences |
| topic | Dynamical Systems 11B30, 37A45 |
| url | https://arxiv.org/abs/1808.05609 |