On recurrence sets for toral endomorphisms

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 ${\mathbb{T}}^2={\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$, induced by $A$. Given $\tau >0$, let $$R_{\tau }=\{x\in {\mathbb{T}}^2:T^{n}x\in B(x,e^{-n\tau })\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{y}n\in \mathbb{N}\}\mathrm{.}$$ We calculated the Hausdorff dimension of $R_{\tau }$, and also prove that $R_{\tau }$ has a large intersection property.