Turkish Journal of Mathematics
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
