Shrinking targets versus recurrence: the quantitative theory

TL;DR

This paper develops a quantitative theory for recurrence in expanding maps, providing precise asymptotics for the number of close returns and extending results to higher dimensions.

Contribution

It introduces a new quantitative estimate for recurrence counts in expanding maps, generalizing previous results and including higher-dimensional cases.

Findings

Asymptotic formula for recurrence counts with error bounds

Almost sure convergence for recurrence statistics

Extension of results to higher-dimensional dynamical systems

Abstract

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 $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[ R(x,N;T,\psi) := \#\{1\leq n\leq N: d(T^n x, x) < \psi(n)\}. \] We show that for any $\varepsilon > 0$ we have \[ R(x,N;T,\psi) = \Psi(N)+O\left(\Psi^{1/2}(N) \ (\log\Psi(N))^{3/2+\varepsilon}\right) \] for $\mu$-almost all $x\in X$ and for all $N\in\mathbb N$, where $\Psi(N):= 2 \sum_{n=1}^N \psi(n)$. We also prove a generalization of this result to higher dimensions.