Badly Approximable and Nonrecurrent Sets for Expanding Markov Maps

Abstract

We consider the asymptotic behaviors of the orbits of an expanding Markov system [Formula: see text], and prove that the badly approximable set [Formula: see text] is of full Hausdorff dimension for any given sequence [Formula: see text]. Consequently, the Hansdorff dimension of the set of nonrecurrent points in the sense that [Formula: see text] is also full. The results can be applied to [Formula: see text]-transformations, Gauss maps and Lüroth maps, etc.