Poincaré's recurrence theorems for set-valued dynamical systems
Given two topological spaces \(X\) and \(Y\), the authors define a `set-valued dynamical system' as a map \(F:X \to 2^Y\). When \(X=Y\), a notion of `an invariant measure' is defined and it is shown that if \(X\) is a metric space and \(F\) is a compact, upper semi-continuous map with non-empty closed values then the system \((X,F)\) admits an `invariant measure'. Given two set-valued dynamical systems \(F_1: X\to 2^Y\) and \(F_2: Y\to 2^X\) (where \(X\) and \(Y\) are metric spaces) the authors define a notion of `a pair of coincident invariant measures' and prove their existence under certain conditions. Furthermore, they formulate and prove a version of Poincaré recurrence result in their set-up.
scientific article; zbMATH DE number 1106436
scientific article