Distributional chaos for strongly continuous semigroups of operators
Angela A. Albanese Xavier Barrachina Elisabetta M. Mangino Alfredo Peris
Communications on Pure & Applied Analysis 2013, 12(5): 2069-2082 doi: 10.3934/cpaa.2013.12.2069
Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(R)$ by a translation of the Ornstein-Uhlenbeck operator is also given.
keywords: Distributional chaos hypercyclic operators irregular vectors criterion for distributional chaos. $C_0$-semigroups
Chaotic translation semigroups
José A. Conejero Alfredo Peris
Conference Publications 2007, 2007(Special): 269-276 doi: 10.3934/proc.2007.2007.269
We characterize chaos for the translation semigroup, with a sector in the complex plane as index set, defined on a weighted function space. The results are stated in terms of the integrability of the weight function, and in terms of the existence of periodic points. We generalize previous results of [8, 15]. Some examples are also provided to complete the study.
keywords: Chaotic semigroups of operators.
Hypercyclic translation $C_0$-semigroups on complex sectors
José A. Conejero Alfredo Peris
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1195-1208 doi: 10.3934/dcds.2009.25.1195
We study the hypercyclic behaviour of sequences of operators in a $C_0$-semigroup whose index set is a sector in the complex plane. The hypercyclicity and chaos for the concrete case of the translation semigroup is analyzed. Some examples are provided to complete the results.
keywords: infinite-dimensional linear systems translation semigroup. topological transitivity topologically mixing Chaotic semigroup hypercyclic semigroup

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