Density properties of orbits for a hypercyclic operator on a Banach space

TL;DR

This paper classifies the density behaviors of orbits for hypercyclic operators on Banach spaces, identifying four distinct cases and providing examples, including weighted shifts and $C_0$-semigroups, to illustrate these phenomena.

Contribution

It introduces a comprehensive classification of orbit density properties for hypercyclic operators, including new examples and extension to $C_0$-semigroups.

Findings

Four distinct density behavior cases for hypercyclic operators.

Existence of examples for each case using weighted backward shifts.

Extension of results to $C_0$-semigroups.

Abstract

We study density properties of orbits for a hypercyclic operator $T$ on a separable Banach space $X$, and show that exactly one of the following four cases holds: (1) every vector in $X$ is asymptotic to zero with density one; (2) generic vectors in $X$ are distributionally irregular of type $1$; (3) generic vectors in $X$ are distributionally irregular of type $2\frac{1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in $X$ is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups.