Hölder Shadowing on Finite Intervals

Sergey Tikhomirov

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Submission date: 19. Nov. 2013 (revised version: November 2013)
Pages: 22
published in: Ergodic theory and dynamical systems, 35 (2015) 6, p. 2000-2016 
DOI number (of the published article): 10.1017/etds.2014.7
Bibtex
MSC-Numbers: 37C50, 37D20
Keywords and phrases: Hoelder shadowing, structural stability, exponential dichotomy, sublinear growth
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Abstract:
For any θ,ω > 1∕2 we prove that, if any d-pseudotrajectory of length 1∕d^ω of a diffeomorphism f ∈ C² can be d^θ-shadowed by an exact trajectory, then f is structurally stable. Previously it was conjectured by Hammel, Yorke and Grebogi 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.