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$f_{\ast }$on$2^{X}$. Our main theme is the interrelation between the dynamics of$f$and$f_{\ast }$. 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$f_{\ast }$induced on${\mathcal{C}}(K,X)$by composition of maps. We mainly study the properties of transitive points of the induced system$(2^{X},f_{\ast })$both topologically and dynamically, and give some examples. We also look into some more properties of the system$(2^{X},f_{\ast })$.