The speed with which an orbit approaches a limit point

Abstract

If f : X → X is a continuous map of a compact metric space,(X,d),x ∈ X and if Θ =(Θ n) is a sequence of positive reals converging to 0, we investigate the properties of the set ω(f,x,Θ)={y ∈ X:d(y,f n(x)) < Θ n infinitely many n ∈ ℕ. We show that ω(f,x,Θ) is a dense G δ subset of for every Θ when x is a recurrent point, even though ω(f,x,Θ) can be disjoint with the orbit of x for some Θ. Under the assumption that f has an invariant non-atomic Borel probability measure μ, 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 μ-almost every y ∈ X, (ii) if μ is ergodic with full support and if D(f)⊂ X is the set of points having dense orbits, then for μ-almost every X ∈ Xand for every y ∈ X/D(f)ω(g,x,Θ) 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 S ⊂ X which is a countable union of Cantor sets such that lim sup n → ∞ d(f fn(x),f sn(y)) > 0 and lim inf n → ∞ d(f fn(x),f sn(y)) > 0 for any two distinct x,y ∈ S and any two distinct g,s ∈ ℕ. 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. © 2012 Copyright Taylor and Francis Group, LLC.