Arithmetic Progressions and Chaos in Linear Dynamics

Abstract

We characterize chaotic linear operators on reflexive Banach spaces in terms of the existence of long arithmetic progressions in the sets of return times. We also show that this characterization does not hold for arbitrary Banach spaces. To achieve this, we study $$\mathcal {F}$$ -hypercyclicity for a family of subsets of the natural numbers associated to the existence of arbitrarily long arithmetic progressions.