Hausdorff dimension of recurrence sets for matrix transformations of tori

Abstract

Let $T:{\mathbb{T}}^d\to {\mathbb{T}}^d$, defined by $Tx=Ax(\mathrm{mod}1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|{\lambda }_1|\le |{\lambda }_2|\le \cdots \le |{\lambda }_d|$. We investigate the Hausdorff dimension of the recurrence set \[R(ψ):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,ψ(n)) {\rm ~for~infinitely~ many~}n\}\] 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 $\alpha =\underset{n\to \infty }{\mathrm{liminf}}\frac{-\log \psi (n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.