Shadowing in CR-Dynamical Systems

Abstract

A CR-dynamical system is a pair $(X,G)$, where $X$ is a compact metric space and $G$ is a closed relation on $X$. We introduce four $(i,j)$-shadowing properties, obtained by independently varying the quantifiers in the definitions of pseudo-orbit and shadowing. These properties coincide with the classical shadowing property for single-valued systems, while $(2,2)$-shadowing agrees with the established notion of shadowing for set-valued dynamical systems. We investigate how the classical characterisations of shadowing for the identity map, periodic maps and isometries extend to closed relations. In particular, we obtain rigidity results for $(2,1)$-shadowing and positive results for $(1,2)$-shadowing on totally disconnected spaces. Finally, whenever the infinite Mahavier product is non-empty, we prove that a closed relation has the $(2,2)$-shadowing property if, and only if, its associated Mahavier dynamical system has the shadowing property. This removes the surjectivity hypothesis from a recent corresponding result for set-valued dynamical systems.