TY - JOUR

T1 - Hölder shadowing on finite intervals

AU - Tikhomirov, Sergey

PY - 2014

Y1 - 2014

N2 - For any $\theta, \omega>1/2$, we prove that, if any $d$-pseudotrajectory of length $∼1/d\omega$ 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 [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 $\theta=\omega=1/2$, 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.

AB - For any $\theta, \omega>1/2$, we prove that, if any $d$-pseudotrajectory of length $∼1/d\omega$ 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 [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 $\theta=\omega=1/2$, 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.

KW - Holder shadowing

KW - exponential dichotomy

KW - sublinear growth property

U2 - 10.1017/etds.2014.7

DO - 10.1017/etds.2014.7

M3 - Article

VL - 35

SP - 2000

EP - 2016

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 6

ER -