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The $C^{0}$ integrability of symplectic twist maps without conjugate points

Part of: Smooth dynamical systems: general theory Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Low-dimensional dynamical systems

Published online by Cambridge University Press:  15 August 2019

MARC ARCOSTANZO
MARC ARCOSTANZO*
Affiliation:
Laboratoire de Mathématiques d’Avignon (EA2151), Avignon Université, France email [email protected]
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Abstract

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is  $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Keywords

symplectic twist mapcomplete integrabilityweak KAM theory

MSC classification

Primary: 37C25: Fixed points, periodic points, fixed-point index theory 37E40: Twist maps 37J35: Completely integrable systems, topological structure of phase space, integration methods
Secondary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 1 , January 2021 , pp. 48 - 65
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Copyright
© Cambridge University Press, 2019

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