Frequently hypercyclic operators with irregularly visiting orbits

TL;DR

This paper demonstrates that certain strongly frequently hypercyclic operators on Banach spaces possess vectors with irregularly visiting orbits, meaning their return times to open sets can have positive lower density but lack a well-defined density.

Contribution

It establishes the existence of vectors with irregularly visiting orbits for operators satisfying a strong form of the Frequent Hypercyclicity Criterion, extending understanding of hypercyclic dynamics.

Findings

Existence of vectors with irregular visiting orbits in strongly frequently hypercyclic operators.

Such vectors have positive lower density of return times to every open set.

There are open sets where the return times lack a density.

Abstract

We prove that a bounded operator $T$ on a separable Banach space $X$ satisfying a strong form of the Frequent Hypercyclicity Criterion (which implies in particular that the operator is universal in the sense of Glasner and Weiss) admits frequently hypercyclic vectors with irregularly visiting orbits, i.e. vectors $x\in X$ such that the set $\mathcal{N}_T(x,U)=\{n\ge 1\,;\,T^{n}x\in U\}$ of return times of $x$ into $U$ under the action of $T$ has positive lower density for every non-empty open set $U\subseteq X$, but there exists a non-empty open set $U_0\subseteq X$ such that $\nt{x}{U_0}$ has no density.