A remark on the frequent hypercyclicity criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation

TL;DR

This paper extends the frequent hypercyclicity criterion to weighted composition semigroups, linking chaos and hypercyclicity, and applies it to the linear von Foerster-Lasota equation in mathematical biology.

Contribution

It generalizes the frequent hypercyclicity criterion from translation semigroups to weighted composition semigroups, with applications to PDEs in biology.

Findings

Equivalence of chaos and the frequent hypercyclicity criterion for certain semigroups.

Application to the linear von Foerster-Lasota equation.

Analysis of weighted composition semigroups on Sobolev spaces.

Abstract

We generalize a result for the translation $C_0$-semigroup on $L^p(\R_+,\mu)$ about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity criterion due to Mangino and Peris to certain weighted composition $C_0$-semigroups. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster-Lasota equation arising in mathematical biology. Weighted composition $C_0$-semigroups on Sobolev spaces are also considered.