Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-18T08:40:01.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Lyapunov exponents for expansive homeomorphisms

Part of: Topological dynamics

Published online by Cambridge University Press:  10 February 2020

M. J. Pacifico  and
J. L. Vieitez
M. J. Pacifico
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, R.J., Brazil (pacifico@im.ufrj.br)
J. L. Vieitez
Affiliation:
Facultad de Ingenieria, Instituto de Matemática, Universidad de la Republica, CC30, CP 11300, Montevideo, Uruguay (jvieitez@fing.edu.uy)
Get access Rights & Permissions [Opens in a new window]

Abstract

We address the problem of defining Lyapunov exponents for an expansive homeomorphism f on a compact metric space (X, dist) using similar techniques as those developed in Barreira and Silva [Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys.13 (2005), 469–490]; Kifer [Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys.3 (1983), 119–127]. Under certain conditions on the topology of the space X where f acts we obtain that there is a metric D defining the topology of X such that the Lyapunov exponents of f are different from zero with respect to D for every point x ∈ X. We give an example showing that this may not be true with respect to the original metric dist. But expansiveness of f ensures that Lyapunov exponents do not vanish on a Gδ subset of X with respect to any metric defining the topology of X. We define Lyapunov exponents on compact invariant sets of Peano spaces and prove that if the maximal exponent on the compact set is negative then the compact is an attractor.

Keywords

expansive homeomorphismsLyapunov exponentsPeano space

MSC classification

Primary: 37B25: Lyapunov functions and stability; attractors, repellers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 413 - 425
Copyright
Copyright © Edinburgh Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Barreira, L. and Silva, C., Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys. 13 (2005), 469–490.CrossRefGoogle Scholar
2.Bessa, M. and Silva, C., Dense area-preserving homeomorphisms have zero Lyapunov exponents, Discrete Contin. Dynam. Sys. 32(4) (2012), 1231–1244.CrossRefGoogle Scholar
3.Durand-Cartagena, E. and Jaramillo, J. A., Pointwise Lipschitz functions on metric spaces, J. Math. Anal. Appl. 363(2) (2010), 525–548.CrossRefGoogle Scholar
4.Fathi, A., Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys. 126 (1989), 249–262.CrossRefGoogle Scholar
5.Kifer, Y., Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys. 3 (1983), 119–127.CrossRefGoogle Scholar
6.Lewowicz, J., Persistence in expansive systems, Ergod. Th. Dynam. Sys. 3 (1983), 567–578.CrossRefGoogle Scholar
7.Mañé, R., Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313–319.CrossRefGoogle Scholar
8.Myers, S. B., Arcs and geodesics in metric spaces, Trans. Amer. Math. Soc. 57(2) (1945), 217–227.CrossRefGoogle Scholar
9.Morales, C. A., Thieullen, P. and Villavicencio, H., Lyapunov exponents on metric spaces, Bull. Aust. Math. Soc. 97 (2018), 153–162.CrossRefGoogle Scholar
10.Sprott, J. C., Numerical Calculation of Largest Lyapunov Exponent. Technical Note, Department of Physics, University of Wisconsin, Madison, WI, USA (2015). Available at http://sprott.physics.wisc.edu/chaos/lyapexp.htmGoogle Scholar