Hausdorff dimension of recurrence sets for matrix transformations of tori

Abstract

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\dots\le|\lambda_d|$. We investigate the Hausdorff dimension of the recurrence set \[R(\psi):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,\psi(n)) {\rm ~for~infinitely~ many~}n\}\] for $\alpha\ge\log|\lambda_d/\lambda_1|$, where $\psi$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $\alpha=\liminf\limits_{n\to\infty}\frac{-\log \psi(n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.