On recurrence sets for toral endomorphisms

Abstract

Let $A$ be a $2\times 2$ integral matrix with an eigenvalue of modulus strictly less than 1. Let $T$ be the natural endomorphism on the torus ${\mathbb{T}}^2={\mathbb{R}}^2/{\mathbb{Z}}^2$, induced by $A$. Given $\tau >0$, let $R_{\tau }$ be the set of points $x\in {\mathbb{T}}^2$ such that $T^{n}x\in B(x,e^{-n\tau })$ for infinitely many $n\in \mathbb{N}$. We obtain a formula for the Hausdorff dimension of $R_{\tau }$, and also prove that $R_{\tau }$ has a large intersection property.