Hausdorff dimension of recurrence sets

Abstract

We consider linear mappings on the 2-dimensional torus, defined by T ( x ) = A x ( mod 1 ) , where A is an invertible 2 × 2 integer matrix, with no eigenvalues on the unit circle. In the case det A = ± 1 , we give a formula for the Hausdorff dimension of the set <?CDATA $ \left\{\, x \in \mathbb{T}^2 : d \left(T^{{\,}n} \left(x\right), x\right) { x ∈ T 2 : d ( T n ( x ) , x ) &lt; e − α n for infinitely many n } .