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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 morales@impa.br
P. THIEULLEN
Affiliation:
Institut de Mathématiques, Université de Bordeaux I, 33405, Talence, France email philippe.thieullen@u-bordeaux.fr
H. VILLAVICENCIO
Affiliation:
Instituto de Matemática y Ciencias Afines, Lima, Perú email hvillavicencio@imca.edu.pe
*
✉morales@impa.br
Rights & Permissions [Opens in a new window]

Abstract

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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.

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