Dynamics of Induced Systems

Abstract

In this paper, we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if $X$ is a metric space, let $2^X$ denote the space of non-empty compact subsets of $X$ provided with the Hausdorff topology. If $f$ is a continuous self-map on $X$, there is a naturally induced continuous self-map $f_*$ on $2^X$. Our main theme is the interrelation between the dynamics of $f$ and $f_*$. For such a study, it is useful to consider the space $\mathcal{C}(K,X)$ of continuous maps from a Cantor set $K$ to $X$ provided with the topology of uniform convergence, and $f_*$ induced on $\mathcal{C}(K,X)$ by composition of maps. We mainly study the properties of transitive points of the induced system $(2^X,f_*)$ both topologically and dynamically, and give some examples. We also look into some more properties of the system $(2^X,f_*)$.