On the Hausdorff dimension of the recurrent sets induced from endomorphisms of free groups

We show that F. Dekking’s recurrent sets in \mathbb{R}^2 , which correspond to Markov partitions for conformally expanding maps of the 2-torus, have Hausdorff dimension strictly greater than one. This is a counterpart to the classical result of R. Bowen on the non-smoothness of the Markov partitions for Anosov diffeomorphisms of the 3-torus.We also present a non-conformal example where the recurrent set is a parallelogram and hence its Hausdorff dimension is one.