On uniform recurrence for hyperbolic automorphisms of the $2$-dimensional torus

We are interested in studying sets of the form $$\mathcal{U}(\alpha ):=\{x\in X:\mathrm{\exists }M=M(x)\ge 1\text{ such that }\mathrm{\forall }N\ge M,\mathrm{\exists }n\le N\text{ such that }d(T^{n}x,x)\le \mathrm{\mid }\lambda {\mathrm{\mid }}^{-\alpha N}\}$$ where $(X,T,d)$ is our metric dynamical system and $\mathrm{\mid }\lambda \mathrm{\mid }>1$. Although a lot of results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider $X={\mathbb{T}}^2$, $T(x)=Ax(\mathrm{m}\mathrm{o}\mathrm{d}1)$, where $A$ is a hyperbolic, area preserving, $2\times 2$ matrix with integer entries and $\lambda$ is the eigenvalue of $A$ of modulus larger than $1$ and we explicitly calculate the Hausdorff dimension of this set.