On eventually always hitting points

TL;DR

This paper studies the behavior of points in dynamical systems with exponential decay of correlations, providing conditions for full measure of eventually always hitting points and asymptotic estimates related to hitting times and continued fraction coefficients.

Contribution

It establishes sufficient conditions for the set of eventually always hitting points to have full measure and derives asymptotic estimates for hitting times in systems with exponential decay.

Findings

Full measure of eventually always hitting points under certain conditions.

Asymptotic estimates for the number of hits in dynamical systems.

Application to continued fraction coefficients of almost every point.

Abstract

We consider dynamical systems $(X,T,\mu)$ which have exponential decay of correlations for either H\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\infty$, we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points $x$ such that for all large enough $m$, there is a $k < m$ with $T^k (x) \in B_m$. We also give an asymptotic estimate as $m \to \infty$ on the number of $k < m$ with $T^k (x) \in B_m$. As an application, we prove for almost every point $x$ an asymptotic estimate on the number of $k \leq m$ such that $a_k \geq m^t$, where $t \in (0,1)$ and $a_k$ are the continued fraction coefficients of $x$.