On transitive and chaotic dynamics of linear semiflows

Abstract

In this paper, we study the underlying relationship between transitivity and chaos for linear semiflows, which provides a better understanding of chaotic dynamics in topological spaces. We show that syndetically transitive linear semiflows with abelian acting semigroups are thickly syndetically-transitive and weakly mixing. As a result, Devaney chaos implies the weak mixing for linear semiflows with abelian acting semigroups. Besides, we give the characterizations for multi-dimensional Li-Yorke chaotic linear semiflows and show that if a linear semiflow (S,X) on a Polish space satisfies one of the following conditions: (i) S is discrete; (ii) dimX=∞; (iii) S is abelian, then Devaney chaos implies multi-dimensional Li-Yorke chaos. This answers a latest question posed in [12] (Question 2.19) to some extent. We also show that any topologically transitive linear semiflow (S,X) on a Polish space X with a translation-invariant metric appears to be densely Li-Yorke chaotic if IntX(Bx)=∅ for any compact subset B of S and any transitive point x∈X. Moreover, if the acting semigroup S is abelian, we present a weaker condition for the densely multi-dimensional Li-Yorke chaos.