On eventually always hitting points

Abstract

We consider dynamical systems $(X,T,\mu )$ which have exponential decay of correlations for either Hölder 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 $k\le m$ such that $a_k\ge m^t$, where $t\in (0,1)$ and $a_k$ are the continued fraction coefficients of $x$.

minor changes. 18 pages