The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents

Abstract

We consider the irrational Aubry-Mather sets of an exact symplectic monotone $C^1$ twist map of the two-dimensional annulus, introduce for them a notion of “$C^1$-regularity” (related to the notion of Bouligand paratingent cone) and prove that

  • $\bullet$  a Mather measure has zero Lyapunov exponents if and only if its support is $C^1$-regular almost everywhere;
  • $\bullet$  a Mather measure has nonzero Lyapunov exponents if and only if its support is $C^1$-irregular almost everywhere;
  • $\bullet$  an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere;
  • $\bullet$  the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be $C^1$-irregular, are not “too irregular” (i.e., have small paratingent cones).

The main tools that we use in the proofs are the so-called Green bundles.

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      AUTHOR = {Ma{ñ}{é},
      Ricardo},
      TITLE = {Quasi-{A}nosov diffeomorphisms and hyperbolic manifolds},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {229},
      YEAR = {1977},
      PAGES = {351--370},
      ISSN = {0002-9947},
      MRCLASS = {58F15},
      MRNUMBER = {0482849},
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      DOI = {10.2307/1998515},
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      VOLUME = {8},
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      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
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      ISBN = {3-540-15278-4},
      MRCLASS = {58F11 (28Dxx)},
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      VOLUME = {21},
      YEAR = {1982},
      NUMBER = {4},
      PAGES = {457--467},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {58F15},
      MRNUMBER = {0670747},
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      JOURNAL = {Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II},
      FJOURNAL = {Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse},
      VOLUME = {1962},
      YEAR = {1962},
      PAGES = {1--20},
      ISSN = {0065-5295},
      MRCLASS = {34.65 (57.48)},
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      AUTHOR = {Pollicott, Mark},
      TITLE = {Lectures on Ergodic Theory and {P}esin Theory on Compact Manifolds},
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      VOLUME = {180},
      PUBLISHER = {Cambridge Univ. Press},
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      PAGES = {x+162},
      ISBN = {0-521-43593-5},
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      AUTHOR = {R{ü}ssmann, Helmut},
      TITLE = {On the existence of invariant curves of twist mappings of an annulus},
      BOOKTITLE = {Geometric Dynamics},
      VENUE={{R}io de {J}aneiro, 1981},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1007},
      PAGES = {677--718},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
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      MRCLASS = {58F05 (58F10)},
      MRNUMBER = {0730294},
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      }

Authors

Marie-Claude Arnaud

Université d'Avignon et des Pays de Vaucluse
33 rue Louis Pasteur
84000 Avignon cedex
France