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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.22993 |
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| _version_ | 1866917823490555904 |
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| author | Levesley, Jason Li, Bing Simmons, David Velani, Sanju |
| author_facet | Levesley, Jason Li, Bing Simmons, David Velani, Sanju |
| contents | Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $ψ:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[ R(x,N;T,ψ) := \#\{1\leq n\leq N: d(T^n x, x) < ψ(n)\}. \] We show that for any $\varepsilon > 0$ we have \[ R(x,N;T,ψ) = Ψ(N)+O\left(Ψ^{1/2}(N) \ (\logΨ(N))^{3/2+\varepsilon}\right) \] for $μ$-almost all $x\in X$ and for all $N\in\mathbb N$, where $Ψ(N):= 2 \sum_{n=1}^N ψ(n)$. We also prove a generalization of this result to higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shrinking targets versus recurrence: the quantitative theory Levesley, Jason Li, Bing Simmons, David Velani, Sanju Dynamical Systems Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $ψ:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[ R(x,N;T,ψ) := \#\{1\leq n\leq N: d(T^n x, x) < ψ(n)\}. \] We show that for any $\varepsilon > 0$ we have \[ R(x,N;T,ψ) = Ψ(N)+O\left(Ψ^{1/2}(N) \ (\logΨ(N))^{3/2+\varepsilon}\right) \] for $μ$-almost all $x\in X$ and for all $N\in\mathbb N$, where $Ψ(N):= 2 \sum_{n=1}^N ψ(n)$. We also prove a generalization of this result to higher dimensions. |
| title | Shrinking targets versus recurrence: the quantitative theory |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.22993 |