Quantitative recurrence properties of expanding maps
Abstract
Under a map T, a point x recurs at rate given by a sequence rn near a point x0 if d(Tn(x),x0)< rn infinitely often. Let us fix x0, and consider the set of those x's. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T, the local dimension near x0 and the upper limit of 1/n log 1/rn. We apply our results in several concrete examples including subshifts of finite type, Gauss transformation and inner functions.
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