Quantitative Recurrence Properties for Systems with Non-uniform Structure

Let X be a subshift with non-uniform structure, and sigma: X -> X be a shift map. Further, define R(psi) := {x is an element of X : d(sigma(n) x,x) < psi (n) for infinitely many n} and R(f):= {x is an element of X : d(sigma(n)x,x) < e(-Snf(x)) for infinitely many n}, where psi: N -> R+ is a nonincreasing and positive function and f : X -> R+ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, dim(H) R(psi) can be expressed by psi and dim(H) R(f) is the solution of the Bowen equation of topological pressure. These results can be applied to a large class of symbolic systems, including beta-shifts, S-gap shifts, and their factors.