Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T14:05:16.741Z Has data issue: false hasContentIssue false
  • English
  • Français

NONUNIFORM EXPONENTIAL BEHAVIOUR AND TOPOLOGICAL EQUIVALENCE

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  21 July 2015

LUIS BARREIRA ,
LIVIU HORIA POPESCU  and
CLAUDIA VALLS
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal e-mail: barreira@math.ist.utl.pt
LIVIU HORIA POPESCU
Affiliation:
Department of Mathematics and Informatics, Oradea University, Str. Universitatii Nr.1, 410087 Oradea, Romania e-mail: lpopescu2002@yahoo.com
CLAUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal e-mail: cvalls@math.ist.utl.pt
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that any evolution family with a strong nonuniform exponential dichotomy can always be transformed by a topological equivalence to a canonical form that contracts and/or expands the same in all directions. We emphasize that strong nonuniform exponential dichotomies are ubiquitous in the context of ergodic theory. The main novelty of our work is that we are able to control the asymptotic behaviour of the topological conjugacies at the origin and at infinity.

MSC classification

Primary: 37D10: Invariant manifold theory
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 279 - 291
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Appell, J., Lakshmikantham, V., Van Minh, N. and Zabreiko, P., A general model of evolutionary processes. Exponential dichotomy. I, II, Nonlinear Anal. 21 (1993), 207–218, 219–225.Google Scholar
2.Barreira, L. and Pesin, Ya., Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115 (Cambridge University Press, Cambridge, 2007).Google Scholar
3.Barreira, L. and Schmeling, J., Sets of “non-typical“ points have full topological entropy and full Hausdorff dimension, Isr. J. Math. 116 (2000), 29–70.CrossRefGoogle Scholar
4.Barreira, L. and Valls, C., Conjugacies for linear and nonlinear perturbations of nonuniform behavior, J. Funct. Anal. 253 (2007), 324–358.Google Scholar
5.Barreira, L. and Valls, C., Stability of nonautonomous differential equations, Lect. Notes. in Math., vol. 1926 (Springer, Berlin, 2008).Google Scholar
6.Hale, J., Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25 (American Mathematical Society, Providence, 1988).Google Scholar
7.Henry, D., Geometric theory of semilinear parabolic equations, Lect. Notes in Math., vol. 840 (Springer, Berlin-New York, 1981).Google Scholar
8.Oseledets, V., A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–221.Google Scholar
9.Palmer, K., A characterization of exponential dichotomy in terms of topological equivalence, J. Math. Anal. Appl. 69 (1979), 8–16.CrossRefGoogle Scholar
10.Palmer, K., The structurally stable linear systems on the half-line are those with exponential dichotomies, J. Differ. Equ. 33 (1979), 16–25.CrossRefGoogle Scholar
11.Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728.Google Scholar
12.Popescu, L. H., A topological classification of linear differential equations on Banach spaces, J. Differ. Equ. 203 (2004), 28–37.CrossRefGoogle Scholar
13.Sell, G. and You, Y., Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143 (Springer, New York, 2002).CrossRefGoogle Scholar
14.Van Minh, N., On a topological classification of nonautonomous differential equations, in Qualitative theory of differential equations (Szeged, 1988), Colloq. Math. Soc. János Bolyai, vol. 53 (North-Holland, 1990), 421–426.Google Scholar