Induced dynamics on the hyperspaces (Q2825441)

Let \((X,d)\) be a compact metric space and let \(F=\{f_1,f_2,\ldots,f_k\}\) be a finite collection of continuous self maps on \(X\). The pair \((X,F)\) generates a multi-valued dynamical system with the rule \(F(x)=\{f_1(x),f_2(x),\ldots,f_k(x) \}\), denoted by \((X,F)\). Now, if \((Y,\tau)\) is a Hausdorff topological space and \(\Psi\) is the subfamily of all non-empty closed subsets of \(Y\), when \(\Psi\) is endowed with the topology \(\Gamma\) generated using the topology \(\tau\) of \(Y\), then the pair \((\Psi,\Gamma)\) is called the hyperspace associated with \((Y,\tau)\).NEWLINENEWLINELet \((X,F)\) be a dynamical system generated by a finite family of continuous self map on \(X\), say \(\{f_1,f_2,\ldots,f_k\}\). A point \(x\) is called periodic if there exists \(n\in \mathbb{N}\) such that \(x\in F^{n}(x)\). The collection \(F\) is transitive if for any pair of non-empty open sets \(U,\) \(V\) in \(X\), there exists \(n\in \mathbb{N}\) such that \(F^n(U)\cap V\neq\emptyset\). The collection \(F\) is super-transitive if for any pair of non-empty open sets \(U,\) \(V\) in \(X\), there exist \(x\in U\) and \(n\in \mathbb{N}\) such that \(F^n(x)\subset V\).NEWLINENEWLINEThe author investigates the relation between the dynamical behavior of \((X,F)\) and the induced system \((\Psi,\overline{F})\). Some interesting result are:NEWLINENEWLINE1. Let \((X,F)\) be a dynamical system generated by a finite commutative family of continuous self-maps on \(X\). If \(F\) is super-transitive, then \(F\) is a singleton.NEWLINENEWLINE2. Let \((X,F)\) be a dynamical system generated by a finite commutative family of continuous self-maps on \(X\). Let \((\Psi,\Delta)\) be the associated hyperspace and let \(\overline{F}\) be the corresponding induced map. If the family \(F\) contains more than one map, then \(\overline{F}\) cannot be transitive.