A strong Borel--Cantelli lemma for recurrence
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914783227281408 |
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| author | Persson, Tomas |
| author_facet | Persson, Tomas |
| contents | Consider a mixing dynamical systems $([0,1], T, μ)$, for instance a piecewise expanding interval map with a Gibbs measure $μ$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $μ(B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to $m_1 + \ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence.
A consequence is that \[
\lim_{r \to 0} \frac{\log τ_{B(x,r)} (x)}{- \log μ(B (x,r))}
= 1 \] for almost every $x$, where $τ$ is the return time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_07344 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A strong Borel--Cantelli lemma for recurrence Persson, Tomas Dynamical Systems 37E05, 37A05, 37B20 Consider a mixing dynamical systems $([0,1], T, μ)$, for instance a piecewise expanding interval map with a Gibbs measure $μ$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $μ(B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to $m_1 + \ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence. A consequence is that \[ \lim_{r \to 0} \frac{\log τ_{B(x,r)} (x)}{- \log μ(B (x,r))} = 1 \] for almost every $x$, where $τ$ is the return time. |
| title | A strong Borel--Cantelli lemma for recurrence |
| topic | Dynamical Systems 37E05, 37A05, 37B20 |
| url | https://arxiv.org/abs/2202.07344 |