Inverse pseudo orbit tracing property for robust diffeomorphisms

Abstract

Let $M$ be a closed smooth Riemannian manifold $M$, and let $f:M\to M$ be a diffeomorphism. Herein, we demonstrate that (i) if $f$ has the $C^1$ robustly inverse shadowing property on the chain recurrent set $\mathcal{C}\mathcal{R}(f)$, then $\mathcal{C}\mathcal{R}(f)$ is hyperbolic and (ii) if $f$ has the $C^1$ robustly inverse shadowing property on a nontrivial transitive set $\Lambda \subset M$, then $\Lambda$ is hyperbolic for $f$. Especially, the item (ii) is a proof of the conjecture of Lee and Lee \cite{LL}.

21 pages