Article contents
Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2
Published online by Cambridge University Press: 20 January 2022
Abstract
The two main results in this paper concern the regularity of the invariant foliation of a
$C^0$
-integrable symplectic twist diffeomorphism of the two-dimensional annulus, namely that (i) the generating function of such a foliation is
$C^1$
, and (ii) the foliation is Hölder with exponent
$\tfrac 12$
. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol’d–Liouville coordinates, in which the dynamics restricted to the leaves is conjugate to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the two-dimensional annulus has Arnol’d–Liouville coordinates and then provide examples of ‘strange’ Lipschitz foliations by smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.
MSC classification
- Type
- Original Article
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- Copyright
- © The Author(s), 2022. Published by Cambridge University Press
References
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