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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.
$C^1$
twist map
We construct a 
$C^1$
symplectic twist map g of the annulus that has an essential invariant curve 
$\Gamma $
such that 
$\Gamma $
is not differentiable and g restricted to 
$\Gamma $
is minimal.
Very few things are known about the curves that are at the boundary of the instability zones of symplectic twist maps. It is known that in general they have an irrational rotation number and that they cannot be KAM curves. We address the following questions. Can they be very smooth? Can they be non-
${C}^{1} $?
Can they have a Diophantine or a Liouville rotation number? We give a partial answer for 
${C}^{1} $ and 
${C}^{2} $ twist maps.
In Theorem 1, we construct a 
${C}^{2} $ symplectic twist map 
$f$ of the annulus that has an essential invariant curve 
$\Gamma $ such that
$\bullet $ 
$\Gamma $ is not differentiable;
$\bullet $ the dynamics of 
${f}_{\vert \Gamma } $ is conjugated to the one of a Denjoy counter-example;
$\bullet $ 
$\Gamma $ is at the boundary of an instability zone for 
$f$.
Using the Hayashi connecting lemma, we prove in Theroem 2 that any symplectic twist map restricted to an essential invariant curve can be embedded as the dynamics along a boundary of an instability zone for some 
${C}^{1} $ symplectic twist map.
