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

Abstract

We are interested in studying sets of the form \[ \mathcal{U}(\alpha) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |\lambda|^{-\alpha N} \right\} \] where $(X,T,d)$ is our metric dynamical system and $|\lambda|>1$. 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 $X=\mathbb{T}^2$, $T(x) = Ax \pmod{1}$, 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.