Skip to main content
×
Home
    • Aa
    • Aa

Hölder shadowing on finite intervals

  • SERGEY TIKHOMIROV (a1) (a2)
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.

Copyright
References
Hide All
L. Barreira and C. Valls . Stable manifolds for nonautonomous equations without exponential dichotomy. J. Differential Equations 221 (2006), 5890.

C. Chicone and Yu Latushkin . Evolution Semigroups in Dynamical Systems and Differential Equations (Mathematical Surveys and Monographs, 70). American Mathematical Society, Providence, RI, 1999.

C. V. Coffman and J. J. Schaeffer . Dichotomies for linear difference equations. Math. Ann. 172 (1967), 139166.

S. M. Hammel , J. A. Yorke and C. Grebogi . Do numerical orbits of chaotic dynamical processes represent true orbits. J. Complexity 3 (1987), 136145.

S. M. Hammel , J. A. Yorke and C. Grebogi . Numerical orbits of chaotic processes represent true orbits. Bull. Amer. Math. Soc. 19 (1988), 465469.

N. T. Huy and N. Van Minh . Exponential dichotomy of difference equations and applications to evolution equations on the half-line. Comput. Math. Appl. 42 (2001), 301311.

R. Mañé . Characterizations of AS diffeomorphisms. Geometry and Topology (Lecture Notes in Mathematics, 597). Springer, Berlin, 1977, pp. 389394.

A. V. Osipov , S. Yu Pilyugin and S. B. Tikhomirov . Periodic shadowing and Ω-stability. Regul. Chaotic Dyn. 15 (2010), 404417.

K. Palmer . Shadowing in Dynamical Systems. Theory and Applications. Kluwer, Dordrecht, 2000.

K. J. Palmer . Exponential dichotomies and transversal homoclinic points. J. Differential Equations 55 (1984), 225256.

K. J. Palmer . Exponential dichotomies and Fredholm operators. Proc. Amer. Math. Soc. 104 (1988), 149156.

K. J. Palmer . Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynam. Rep. 1 (1988), 265306.

S. Yu Pilyugin . Generalizations of the notion of hyperbolicity. J. Difference Equ. Appl. 12 (2006), 271282.

S. Yu Pilyugin . Variational shadowing. Discrete Contin. Dyn. Syst. Ser. B 14 (2010), 733737.

S. Yu Pilyugin and S. B. Tikhomirov . Lipschitz Shadowing implies structural stability. Nonlinearity 23 (2010), 25092515.

S. Yu Pilyugin . Theory of pseudo-orbit shadowing in dynamical systems. Differ. Equ. 47 (2011), 19291938.

C. Robinson . Stability theorems and hyperbolicity in dynamical systems. Rocky Mountain J. Math. 7 (1977), 425437.

V. Slyusarchuk . Exponential dichotomy of solutions of discrete systems. Ukrain. Mat. Zh. 35 (1983), 109115.

D. Todorov . Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory. Discrete Contin. Dyn. Syst. Ser. A 33 (2013), 41874205.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 75 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th October 2017. This data will be updated every 24 hours.