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Construction of invariant measures supported within the gaps of Aubry–Mather sets

Published online by Cambridge University Press:  19 September 2008

Giovanni Forni
Giovanni Forni
Affiliation:
Dipartimento di Matematica, Università di Bologna, 40127 Bologna, Italy
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Abstract

This paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 1 , February 1996 , pp. 51 - 86
Copyright
Copyright © Cambridge University Press 1996

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