Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T17:06:20.637Z Has data issue: false hasContentIssue false
  • English
  • Français

LYAPUNOV EXPONENTS ON METRIC SPACES

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  04 October 2017

C. A. MORALES Open the ORCID record for C. A. MORALES [Opens in a new window] ,
P. THIEULLEN  and
H. VILLAVICENCIO
C. A. MORALES*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970 Rio de Janeiro, Brazil email [email protected]
P. THIEULLEN
Affiliation:
Institut de Mathématiques, Université de Bordeaux I, 33405, Talence, France email [email protected]
H. VILLAVICENCIO
Affiliation:
Instituto de Matemática y Ciencias Afines, Lima, Perú email [email protected]
*
✉[email protected]
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 use the pointwise Lipschitz constant to define an upper Lyapunov exponent for maps on metric spaces different to that given by Kifer [‘Characteristic exponents of dynamical systems in metric spaces’, Ergodic Theory Dynam. Systems3(1) (1983), 119–127]. We prove that this exponent reduces to that of Bessa and Silva on Riemannian manifolds and is not larger than that of Kifer at stable points. We also prove that it is invariant along orbits in the case of (topological) diffeomorphisms and under topological conjugacy. Moreover, the periodic orbits where this exponent is negative are asymptotically stable. Finally, we estimate this exponent for certain hyperbolic homeomorphisms.

Keywords

upper Lyapunov exponentmetric spacepointwise Lipschitz constant

MSC classification

Primary: 54H20: Topological dynamics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 153 - 162
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was partially supported by CNPq from Brazil, the third author was partially supported by FONDECYT from Peru (C.G. 217–2014); the work was also partially supported by MATHAMSUB 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

References

Arnold, V. I., Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60 (Springer, New York–Heidelberg, 1978).Google Scholar
Barreira, L. and Silva, C. M., ‘Lyapunov exponents for continuous transformations and dimension theory’, Discrete Contin. Dyn. Syst. 13(2) (2005), 469–490.CrossRefGoogle Scholar
Bessa, M. and Silva, C. M., ‘Dense area-preserving homeomorphisms have zero Lyapunov exponents’, Discrete Contin. Dyn. Syst. 32(4) (2012), 1231–1244.Google Scholar
Bylov, D., Vinograd, R., Grobman, D. and Nemyckii, V., Theory of Lyapunov Exponents and its Application to Problems of Stability (Nauka, Moscow, 1966) (in Russian).Google Scholar
Charatonik, W. J. and Insall, M., ‘Absolute differentiation in metric spaces’, Houston J. Math. 38(4) (2012), 1313–1328.Google Scholar
Durand-Cartagena, E. and Jaramillo, J. A., ‘Pointwise Lipschitz functions on metric spaces’, J. Math. Anal. Appl. 363(2) (2010), 525–548.Google Scholar
Fathi, A., ‘Expansiveness, hyperbolicity and Hausdorff dimension’, Comm. Math. Phys. 126(2) (1989), 249–262.Google Scholar
Kifer, Y., ‘Characteristic exponents of dynamical systems in metric spaces’, Ergodic Theory Dynam. Systems 3(1) (1983), 119–127.Google Scholar
Kloeden, P. E. and Ombach, J., ‘Hyperbolic homeomorphisms and bishadowing’, Ann. Polon. Math. 65(2) (1997), 171–177.CrossRefGoogle Scholar
Mañé, R., Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8 (Springer, Berlin, 1987), translated from the Portuguese by Silvio Levy.Google Scholar
Ombach, J., ‘Shadowing, expansiveness and hyperbolic homeomorphisms’, J. Aust. Math. Soc. Ser. A 61(1) (1996), 57–72.Google Scholar
Park, J. S., Lee, K. and Koo, K.-S., ‘Hyperbolic homeomorphisms’, Bull. Korean Math. Soc. 32(1) (1995), 93–102.Google Scholar
Reddy, W. L., ‘Expanding maps on compact metric spaces’, Topology Appl. 13(3) (1982), 327–334.Google Scholar
Sakai, K., ‘Shadowing properties of L-hyperbolic homeomorphisms’, Topology Appl. 112(3) (2001), 229–243.Google Scholar