Holder Shadowing on Finite Intervals

TL;DR

The paper establishes that certain shadowing properties of pseudotrajectories imply the structural stability of diffeomorphisms, extending conjectures for non-uniform hyperbolic systems.

Contribution

It proves that specific shadowing conditions on pseudotrajectories imply structural stability, introducing the sublinear growth property and its link to exponential dichotomy.

Findings

Shadowing property implies structural stability.

Introduction of sublinear growth property for linear equations.

Proof that sublinear growth implies exponential dichotomy.

Abstract

For any $\theta, \omega > 1/2$ we prove that, if any $d$-pseudotrajectory of length $\sim 1/d^{\omega}$ of a diffeomorphism $f\in C^2$ can be $d^{\theta}$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured by Hammel-Grebogi-Yorke that for $\theta = \omega = 1/2$ this property holds for a wide class of non-uniformly hyperbolic diffeomorphisms. In the proof we introduce the notion of sublinear growth property for inhomogenious linear equations and prove that it implies exponential dichotomy.