Article Title

A note on dynamics in functional spaces

Authors

LIANGXUE PENG
CHONG YANG

DOI

10.3906/mat-1512-40

Abstract

In this note, we study topologically transitive and hypercyclic composition operators on $C_p(X)$ or $C_k(X)$. We prove that if $G$ is a semigroup of continuous self maps of a countable metric space $X$ with the following properties: (1) every element of $G$ is one-to-one on $X$, (2) the action of $G$ is strongly run-away on $X$, then the action of $\hat{G}$ on $C_p(X)$ is topologically transitive and hypercyclic. If $G$ is the set of all one-to-one and continuous self maps of $\mathbb{R}\setminus \mathbb{Z}$, then the action of $\hat{G}$ on $C_k(\mathbb{R}\setminus \mathbb{Z})$ is hypercyclic. We also show that the action of $\hat{G}$ on $C_p(\omega_1)$ is not hypercyclic.

Keywords

Topological transitivity, hypercyclicity, composition operators, semigroup actions

First Page

186

Last Page

192

Recommended Citation

PENG, LIANGXUE and YANG, CHONG (2017) "A note on dynamics in functional spaces," Turkish Journal of Mathematics: Vol. 41: No. 1, Article 17. https://doi.org/10.3906/mat-1512-40
Available at: https://journals.tubitak.gov.tr/math/vol41/iss1/17