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Hölder shadowing on finite intervals


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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
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