Arithmetic progressions and chaos in linear dynamics

TL;DR

This paper characterizes chaotic linear operators on reflexive Banach spaces by linking chaos to the presence of long arithmetic progressions in return times, exploring their relation with hypercyclicity and linear dynamics.

Contribution

It introduces a novel characterization of chaos in linear operators through arithmetic progressions and studies their connection with hypercyclicity in Banach spaces.

Findings

Chaotic operators are characterized by long arithmetic progressions in return times.

The study links $ ext{F}$-hypercyclicity with arithmetic progressions.

Connections between chaos, hypercyclicity, and arithmetic progressions are established.

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. To achieve this, we study $\mathcal F$-hypercyclicity for a family of subsets of the natural numbers associated with the existence of arbitrarily long arithmetic progressions. We investigate their connection with different concepts in linear dynamics.