Dynamics of Induced Systems

TL;DR

This paper explores the relationship between the dynamics of a continuous map on a metric space and the induced map on the space of compact subsets, focusing on transitivity and other properties.

Contribution

It introduces a detailed analysis of the dynamical properties of induced systems on compact subsets, linking the dynamics of the original map to the induced map.

Findings

Characterization of transitive points in the induced system

Examples illustrating the dynamics of the induced system

Insights into the properties of the induced system $(2^X,f_*)$

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 $(2^X,f_*)$.