Diophantine approximation of the orbits in topological dynamical systems

Abstract

We would like to present a general principle for the shrinking target problem in a topological dynamical system. More precisely, let (X, d) be a compact metric space and T : X → X a continuous transformation on X. For any integer valued sequence {an} and y ∈ X, define Ey({an}) = δ>0 x ∈ X : T n x ∈ Ba n (y, δ), for infinitely often n ∈ N , the set of points whose orbit can well approximate a given point infinitely often, where Bn(x, r) denotes the Bowen-ball. It is shown that htop(Ey({an}), T) = 1 1 + a htop(X, T), with a = lim inf n→∞ an n , if the system (X, T) has the specification property. Here htop denotes the topological entropy. An example is also given to indicate that the specification property required in the above result cannot be weakened even to almost specification.