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 $\ddot{I}„>0$, let \[ R_τ=\{\, x\in \mathbb{T}^2 : T^nx\in B(x,e^{-nτ})~\mathrm{infinitely ~many}~n\in\mathbb{N} \,\}. \] We calculated the Hausdorff dimension of $R_{\ddot{I}}„$, and also prove that $R_{\ddot{I}}„$ has a large intersection property.

25 pages, 2 figures