A Simple Characterization of Chaos for Weighted Composition C0-semigroups on Lebesgue and Sobolev Spaces

The purpose of this article is to give a simple characterization of chaos for certain weighted composition C0-semigroups on Lebesgue spaces and Sobolev spaces over open intervals. Recall that a C0-semigroup T on a separable Banach space X is called chaotic if T is hypercyclic, i.e. there is x ∈ X such that {T (t)x; t ≥ 0} is dense in X , and if the set of periodic points, i.e. {x ∈ X ; ∃t > 0 : T (t)x = x}, is dense in X . The study of chaotic C0-semigroups has attracted the attention of many researchers. We refer the reader to Chapter 7 of the monograph by Grosse-Erdmann and Peris [9] and the references therein. Some recent papers in the topic are [1, 4, 5, 8, 14]. For Ω ⊆ R open and a Borel measure μ on Ω admitting a strictly positive Lebesgue density ρ we consider C0-semigroups T on L (Ω, μ), 1 ≤ p < ∞, of the form T (t)f(x) = ht(x)f(φ(t, x)),