标题： 关于环面$2$维超bolic自同构的均匀递归性 显示英文标题

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

摘要： 我们感兴趣的研究对象是形如\[ \mathcal{U}(\alpha) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |\lambda|^{-\alpha N} \right\} \]的集合，其中$(X,T,d)$是我们的度量动力系统，$|\lambda|>1$。尽管对于一维情况已有许多结果，但对高维系统，尤其是双曲情况下的结果则少得多。 我们考虑$X=\mathbb{T}^2$，$T(x) = Ax \pmod{1}$，其中$A$是一个双曲的，保面积的，$2\times 2$矩阵，具有整数条目，$\lambda$是$A$的模大于$1$的特征值，我们显式计算这个集合的豪斯多夫维数。 显示英文摘要

摘要： 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.