Hausdorff dimension of recurrence sets

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