### Area preserving group actions on surfaces.

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We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation...

We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence...

For a positive integer n and R>0, we set $${B}_{R}^{n}=\left\{x\in {\mathbb{R}}^{n}|{\u2225x\u2225}_{\infty}<R\right\}$$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $$h\left(r\right)={\textstyle \frac{1}{2}}{r}_{1}^{2}+...{\textstyle \frac{1}{2}}{r}_{n-1}^{2}+{r}_{n}$$ on $${\mathbb{T}}^{n}\times {B}_{R}^{n}$$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $${\mathbb{T}}^{n}\times {B}_{R}^{n}$$ , and setting $${\epsilon}_{j}:={\u2225h-{H}_{j}\u2225}_{{C}^{0}\left(V\right)}$$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface,...

The main aim of this paper is to give some counterexamples to global invertibility of local diffeomorphisms which are interesting in mechanics. The first is a locally strictly convex function whose gradient is non-injective. The interest in this function is related to the Legendre transform. Then I show two non-injective canonical local diffeomorphisms which are rational: the first is very simple and related to the complex cube, the second is defined on the whole ℝ⁴ and is obtained from a recent...

We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns–Wilkinson and Avila–Santamaria–Viana. Combining this new technique with other constructions we prove that ${C}^{1}$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.

A planar polygonal billiard $\mathcal{P}$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $\mathcal{P}$ there exists a finite number of “blocking” points ${B}_{1},\cdots ,{B}_{n}$ such that every billiard trajectory from $O$ to $A$ meets one of the ${B}_{i}$’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces....

We give an explicit construction of the trace on the algebra of quantum observables on a symplectiv orbifold and propose an index formula.