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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MRREVIEWER = {H. Ehrmann},
ZBLNUMBER = {0107.29301},
} -
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@book {Po, MRKEY = {1215938},
AUTHOR = {Pollicott, Mark},
TITLE = {Lectures on Ergodic Theory and {P}esin Theory on Compact Manifolds},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {180},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1993},
PAGES = {x+162},
ISBN = {0-521-43593-5},
MRCLASS = {58F11},
MRNUMBER = {1215938},
MRREVIEWER = {Nicola{\u\i} T. A. Haydn},
ZBLNUMBER = {0772.58001},
} -
[Ru] H. Rüssmann, "On the existence of invariant curves of twist mappings of an annulus," in Geometric Dynamics, New York: Springer-Verlag, 1983, vol. 1007, pp. 677-718.
@incollection {Ru, MRKEY = {0730294},
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.},
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PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1983},
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MRNUMBER = {0730294},
MRREVIEWER = {Dietrich Flockerzi},
DOI = {10.1007/BFb0061441},
ZBLNUMBER={0531.58041},
} -
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@book {Shu, MRKEY = {0513592},
AUTHOR = {Shub, Michael},
TITLE = {Stabilité Globale des Systèmes Dynamiques},
SERIES = {Astérisque},
VOLUME = {56},
NOTE = {with an English preface and summary},
PUBLISHER = {Société Mathématique de France},
ADDRESS = {Paris},
YEAR = {1978},
PAGES = {iv+211},
MRCLASS = {58F10 (34C35 58F20)},
MRNUMBER = {0513592},
MRREVIEWER = {John Franke},
ZBLNUMBER = {0396.58014},
} -
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@book {Sib, MRKEY = {2076302},
AUTHOR = {Siburg, Karl Friedrich},
TITLE = {The principle of Least Action in Geometry and Dynamics},
SERIES = {Lecture Notes in Math.},
VOLUME = {1844},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
PAGES = {xii+128},
ISBN = {3-540-21944-7},
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MRNUMBER = {2076302},
MRREVIEWER = {Renato Iturriaga},
ZBLNUMBER={1060.37048},
}
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