My co-author George Costakis and I have recently uploaded to arxiv our paper “Szemerédi’s theorem, frequent hypercyclicity and multiple recurrence”. As I’m invited to talk about this subject next month, I will try to give here a general overview of the paper, the notions therein and the main ideas involved in the proofs. Our main objective in this paper is to relate some notions in linear dynamics to more classical notions from topological dynamics. In particular we show that frequently Cesàro hypercyclic operators are necessarily topologically multiply recurrent. The main tool we use to prove this result is Szemerédi’s theorem on arithmetic progressions in sets of positive density. In order to motivate this theorem, I will have to define many standard notions from linear dynamics as well as corresponding notions from topological dynamics. Before discussing the main result and (some of) its applications, I will try to give a picture of hypercyclic operators and their properties, as well as examples of `natural’ operators which are hypercyclic.

** — 1. Introduction: notions of hypercyclicity — **

First of all, I will review some basic notions from linear dynamics that will be quite central throughout the exposition. I refer the reader to the excellent book of Bayart and Matheron (Bayart and Matheron, 2009) where most of this material is drawn from anyways. We will state several classical results here omitting the proof. If no other reference is given, this means the proof can be found in (Bayart and Matheron, 2009).

** — 1.1. Hypercyclic operators — **

We will work on *a separable Banach space* over or . We will always use the symbol to denote a *bounded linear operator* acting on . In what follows I will just write , , without any further comment, assuming always that these symbols have the meaning described above.

The most central notion in linear dynamics is that of hypercyclicity. Continue reading →