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.

Nous montrons des résultats d’existence de points fixes communs pour des homéomorphismes du plan ${ \mathbb{R} }^{2} $ ou la sphère ${ \mathbb{S} }^{2} $, qui commutent deux à deux et préservent une mesure de probabilité. Par exemple, nous montrons que des ${C}^{1} $-difféomorphismes ${f}_{1} , \ldots , {f}_{n} $ de ${ \mathbb{S} }^{2} $ suffisamment proches de l’identité, qui commutent deux à deux, et qui préservent une mesure de probabilité dont le support n’est pas réduit à un point, ont au moins deux points fixes communs.

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

We prove that if f:I=[0,1]→I is a C3-map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R(f) has full Lebesgue measure λ; (ii) both S(f) and Scramb(f) have positive measure. Here R(f), S(f), and Scramb(f) respectively stand for the set of approximately periodic points of f, the set of sensitive points to the initial conditions of f, and the two-dimensional set of points (x,y) such that {x,y} is a scrambled set for f.Also, we show that if f is piecewise monotone and has no wandering intervals, then either λ(R(f))=1 or λ(S(f))>0, and provide examples of maps g,h of this type satisfying S(g)=S(h)=I such that, on the one hand, λ(R(g))=0and λ2 (Scramb (g))=0 , and, on the other hand, λ(R(h))=1 .