Hypercyclic composition operators on spaces of real analytic functions

Abstract

AbstractWe study the dynamical behaviour of composition operators Cϕ defined on spaces (Ω) of real analytic functions on an open subset Ω of ℝd. We characterize when such operators are topologically transitive, i.e. when for every pair of non-empty open sets there is an orbit intersecting both of them. Moreover, under mild assumptions on the composition operator, we investigate when it is sequentially hypercyclic, i.e., when it has a sequentially dense orbit. If ϕ is a self map on a simply connected complex neighbourhood U of ℝ, U ≠ ℂ, then topological transitivity, hypercyclicity and sequential hypercyclicity of Cϕ:(ℝ) → (ℝ) are equivalent.