Mather theory and symplectic rigidity

Mads R. Bisgaard;

doi:10.3934/jmd.2019018

Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations

Communications on Pure & Applied Analysis ◽

10.3934/cpaa.2009.8.683 ◽

2009 ◽

Vol 8 (2) ◽

pp. 683-688

Author(s):

Yasuhiro Fujita ◽

◽

Katsushi Ohmori ◽

Keyword(s):

Mather Theory ◽

Jacobi Equations

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008713 ◽

1996 ◽

Vol 16 (1) ◽

pp. 51-86 ◽

Cited By ~ 3

Author(s):

Giovanni Forni

Keyword(s):

Variational Approach ◽

Invariant Measures ◽

Positive Entropy ◽

Almost Periodic ◽

Invariant Curves ◽

Mather Theory ◽

Entropy Measures ◽

Twist Maps ◽

Minimization Procedure ◽

Area Preserving

AbstractThis 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.

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Chapter Two. From KAM Theory to Aubry-Mather Theory

Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory ◽

10.1515/9781400866618-003 ◽

2015 ◽

pp. 8-17

Keyword(s):

Kam Theory ◽

Mather Theory

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Optimal results in Lorentzian Aubry–Mather theory

manuscripta mathematica ◽

10.1007/s00229-020-01267-2 ◽

2020 ◽

Author(s):

Stefan Suhr

Keyword(s):

Fixed Point ◽

Invariant Measures ◽

Lipschitz Continuity ◽

Fixed Point Theory ◽

Point Theory ◽

Multiplicity Results ◽

Time Separation ◽

Abelian Cover ◽

Mather Theory ◽

Maximal Invariant

AbstractThis article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.

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