Hausdorff dimension of recurrence sets

Abstract

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a formula for the Hausdorff dimension of the set \[ \{ \, x \in \mathbb{T}^d : d (T^n (x), x) < e^{- \alpha n} \text{ for infinitely many } n \, \}. \]