Chaotic Behaviour on Invariant Sets of Linear Operators

Abstract

We study hypercyclicity, Devaney chaos, topological mixing properties and strong mixing in the measure-theoretic sense for opera- tors on topological vector spaces with invariant sets. More precisely, our purpose is to establish links between the fact of satisfying any of our dynamical properties on certain invariant sets, and the corresponding property on the closed linear span of the invariant set, or on the union of the invariant sets. Viceversa, we give conditions on the operator (or C0-semigroup) to ensure that, when restricted to the invariant set, it satis es certain dynamical property. Particular attention is given to the case of positive operators and semigroups on lattices, and the (invariant) positive cone. We also present examples that illustrate these results.