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Aubry–Mather theory for homeomorphisms

Published online by Cambridge University Press:  05 August 2014

ALBERT FATHI  and
PIERRE PAGEAULT
ALBERT FATHI
Affiliation:
Institut universitaire de France, Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France email albert. fathi@ens-lyon. fr
PIERRE PAGEAULT
Affiliation:
Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France email ppageault@gmail. com
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Abstract

In this paper, we develop a variational approach to study the dynamic of a homeomorphism on a compact metric space. In particular, we describe orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semi-distance. We give the link with Auslander’s notion of generalized recurrence, and recover in a different way some parts of a more recent work of Akin and Auslander.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1187 - 1207
Copyright
© Cambridge University Press, 2014 

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References

Akin, E. . The general Topology of Dynamical Systems (Graduate Studies in Mathematics, 1). American Mathematical Society, Providence, RI, 1993.Google Scholar
Akin, E. and Auslander, J. . Generalized recurrence, compactifications, and the Lyapunov topology. Studia Math. 201(1) (2010), 49–63.CrossRefGoogle Scholar
Auslander, J. . Generalized recurrence in dynamical systems. Contributions to Differential Equations 3 (1964), 65–74.Google Scholar
Conley, C. . Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics, 38). American Mathematical Society, Providence, RI, 1978.CrossRefGoogle Scholar
Conley, C. . The gradient structure of a flow. I. Ergodic Theory Dynam. Systems 8* (1988), Charles Conley Memorial Issue, 11–26, 9, with a comment by R. Moeckel.Google Scholar
Hurewicz, W. and Wallman, H. . Dimension Theory (Princeton Mathematical Series, 4). Princeton University Press, Princeton, NJ, 1941.Google Scholar
Mather, J. N. . Variational construction of connecting orbits. Ann. Inst. Fourier 43(5) (1993), 1349–1386.CrossRefGoogle Scholar
Oxtoby, J. C. . Diameters of arcs and the gerrymandering problem. Amer. Math. Monthly 84(3) (1977), 155–162.CrossRefGoogle Scholar
Pageault, P. . Conley barriers and their applications: chain-recurrence and Lyapunov functions. Topology Appl. 156(15) (2009), 2426–2442.CrossRefGoogle Scholar
Shub, M. . Global Stability of Dynamical Systems. Springer, New York, 1987, With the collaboration of Albert Fathi and Rémi Langevin; translated from the French by Joseph Christy.CrossRefGoogle Scholar