The speed with which an orbit approaches a limit point

Abstract

If is a continuous map of a compact metric space , and if is a sequence of positive reals converging to 0, we investigate the properties of the set . We show that is a dense subset of for every when x is a recurrent point, even though can be disjoint with the orbit of x for some . Under the assumption that f has an invariant non-atomic Borel probability measure , we prove results to the effect that (i) there is a uniform upper limit to the speed with which the orbit of each x can approach y for -almost every , (ii) if is ergodic with full support and if is the set of points having dense orbits, then for -almost every and for every there is a uniform upper limit to the speed with which the orbit of x can approach y. Next, using as a useful tool in proofs, we establish the following. If f is totally transitive and X is infinite, then there is a dense subset which is a countable union of Cantor sets such that and for any two distinct and any two distinct . If f is a transitive map enjoying a certain type of continuity in the backward direction, then f has a residual set of points with dense backward orbits.