Shadowing property and entropy on hyperspace of continua induced by Morse gradient system

Abstract: It is known that Morse-Smale diffeomorpshisms 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 subcontiuna does not have the shadowing property. Besides, we provide a simple criterion for $f$ that implies $C(f)$ has infinite topological entropy.