Holder Shadowing on Finite Intervals

For any $θ, ω> 1/2$ we prove that, if any $d$-pseudotrajectory of length $\sim 1/d^ω$ of a diffeomorphism $f\in C^2$ can be $d^θ$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured by Hammel-Grebogi-Yorke that for $θ= ω= 1/2$ 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.

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