Hölder shadowing on finite intervals

Abstract

For any ${\it\theta},{\it\omega}>1/2$, we prove that, if any $d$-pseudotrajectory of length ${\sim}1/d^{{\it\omega}}$ of a diffeomorphism $f\in C^{2}$ can be $d^{{\it\theta}}$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured [S. M. Hammel et al. Do numerical orbits of chaotic dynamical processes represent true orbits. J. Complexity3 (1987), 136–145; S. M. Hammel et al. Numerical orbits of chaotic processes represent true orbits. Bull. Amer. Math. Soc.19 (1988), 465–469] that for ${\it\theta}={\it\omega}=1/2$, this property holds for a wide class of non-uniformly hyperbolic diffeomorphisms. In the proof, we introduce the notion of a sublinear growth property for inhomogeneous linear equations and prove that it implies exponential dichotomy.