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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 x ∈ T 2: d (T n (x), x) e − α n for infinitely many n.
Hu et al. (Tue,) studied this question.