# Inverse pseudo orbit tracing property for robust diffeomorphisms

## TL;DR

This paper proves that robust inverse shadowing properties imply hyperbolicity of chain recurrent and transitive sets in diffeomorphisms, confirming a conjecture and advancing understanding of dynamical stability.

## Contribution

It establishes that $C^1$ robust inverse shadowing on certain sets guarantees their hyperbolicity, confirming a conjecture by Lee and Lee.

## Key findings

- Robust inverse shadowing implies hyperbolicity of chain recurrent sets.
- Robust inverse shadowing on transitive sets implies their hyperbolicity.
- Confirms Lee and Lee's conjecture on hyperbolicity under inverse shadowing.

## 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 $\mathcal{CR}(f)$, then $\mathcal{CR}(f)$ 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}.