Sets That Force Recurrence

We characterize those subsets S of the positive integers with the property that, whenever a point x in a dynamical system enters a compact set K along S, K contains a recurrent point. We do the same for uniform recurrence. Let X denote a (compact) metric space and f : X → X a continuous map. We will use the term (compact) dynamical system to refer to such a pair (X, f) . As usual, we write f for the n-fold composition of f with itself and N for the set of all positive integers. Let R (f) denote the set of recurrent points of f. That is, R (f) = {x ∈ X | for some sequence of integers nk →∞, fk (x)→ x} . A well-known theorem (whose origin we have been unable to determine) states the following: Theorem 1. If (X, f) is a compact dynamical system and U an open set containing R (f) , then for all x ∈ X,