Hausdorff dimension of recurrence sets

Abstract

We consider linear mappings on the $d$-dimensional torus, defined by $T(x)=Ax(\mathrm{mod}1)$, 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 \[ \{ \, x \in \mathbb{T}^d : d (T^n (x), x) < e^{- αn} \text{ for infinitely many } n \, \}. \]

21 pages, 1 figure