Hausdorff dimension of recurrence sets

We consider linear mappings on the 2-dimensional torus, defined by <?CDATA T(x) = Ax \ (mod\ 1)?> T ( x ) = A x ( mod 1 ) , where A is an invertible <?CDATA 2× 2?> 2 × 2 integer matrix, with no eigenvalues on the unit circle. In the case <?CDATA A = ± 1?> det A = ± 1 , we give a formula for the Hausdorff dimension of the set <?CDATA \\, x ∈ T² : d (T^\,n (x), x) < e- α n for infinitely many n \, \.?> { x ∈ T 2 : d ( T n ( x ) , x ) < e − α n for infinitely many n } .