Shadowing property on hyperspace of continua induced by Morse gradient system

Abstract

It is known that Morse-Smale diffeomorphisms have the shadowing property; however, the question of whether $C(f)$ also has the shadowing property when $f$ is Morse-Smale remains open and has been resolved only in a few specific cases~\cite{AB}. We prove that if $f:M\to M$ is a time-one-map of Morse gradient flow, the induced map $C(f):C(M)\to C(M)$ on the hyperspace of subcontinua does not have the shadowing property.