Functional envelopes relative to the point-open topology on a subset

Abstract

If $(X, f)$ is a dynamical system given by a locally compact separable metric space $X$ without isolated points and a continuous map $f : X\to X $ , and $A$ is a countable dense subset of $X$ , then by the functional envelope of $(X, f)$ relative to $\mathcal{P}_A$ we mean the dynamical system $(S_A(X), F_f)$ whose phase space $S_A(X)$ is the space of all continuous selfmaps of $X$ endowed with the point-open topology on $A$ and the map $F_f : S_A(X)\to S_A(X)$ is defined by $F_f (\varphi)=fo\varphi$ for any $\varphi∈ S_A(X)$ . In this paper, we mainly deal with the connection between the properties of a system and the properties of its functional envelope. We show that:(1) $(X, f)$ is weakly mixing if and only if there exists a countable dense subset $A$ of $X$ so that $\big(S_A(X), F_f\big)$ has a transitive point $φ∈ S(X)$ which is surjective; (2) $(X, f)$ is sensitive if and only if $\big(S_A(X), F_f\big)$ is sensitive for every countable dense subset $A$ of $X$ . Moreover, if $(X, f)$ is weakly mixing, then $\big(S_A(X), F_f\big)$ is Auslander-Yorke chaotic for many countable dense subsets $A$ of $X$ . As an application, we consider a class of one-dimensional wave equations with van der Pol boundary condition and show that if the boundary condition is weakly mixing, then there exists an initial condition such that the solutions of the equations exhibit complicated behaviours.