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Shadowing property on hyperspace of continua induced by Morse gradient system

Jelena Katić, Darko Milinković

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.

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 also has the shadowing property when is Morse-Smale remains open and has been resolved only in a few specific cases~\cite{AB}. We prove that if is a time-one-map of Morse gradient flow, the induced map on the hyperspace of subcontinua does not have the shadowing property.

Paper Structure

This paper contains 6 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shadowing
  4. Hyperspaces and induced maps
  5. Morse gradient systems
  6. Proof of Theorem \ref{['thrm:main']}

Key Result

Theorem 1

For any time-one map $f$ of a negative gradient flow of a Morse function on a closed smooth manifold $M$ that satisfies the Morse-Smale condition, the induced homeomorphism $C(f)$ does not satisfy the shadowing property.∎

Theorems & Definitions (6)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Lemma 6