Abstract
The notion of distributional chaos has been recently added to the study of the linear dynamics of operators and C0‐semigroups of operators. We will study this notion of chaos for some examples of C0‐semigroups that are already known to be Devaney chaotic.
Highlights
During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others
Our goal is to study distributional chaos for some C0-semigroups that are already known to be Devaney chaotic
The dynamics exhibited by these C0-semigroups will motivate us to pose some open questions
Summary
Several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. Other definitions of chaos, such as the one introduced by Li-Yorke and the one of distributional chaos introduced by Schweizer and Smıtal, have been considered The relationships between these two notions in the Banach space setting have been recently studied in 3. We recall that a C0-semigroup T is said to be Li-Yorke chaotic if there exists an uncountable subset Γ ⊂ X, called the scrambled set, such that for every pair x, y ∈ Γ of distinct points, we have that lim inf t→∞. The dynamics exhibited by these C0-semigroups will motivate us to pose some open questions
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