We use the properties of $Mp\left(n\right)$ to construct functions ${\mu }_{\ell }:Mp\left(n\right)\to {\mathbf{Z}}_8$ associated with the elements $\ell$ of the lagrangian grassmannian $\Lambda$ (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between $Mp\left(n\right)$ and a subset of $Sp\left(n\right)\times {\mathbf{Z}}_8$, equipped with appropriate algebraic and topological structures.

Soient $\Gamma$ un groupe discret géométriquement fini d’isométries d’une variété de Hadamard pincée $X$ et $𝒫$ une pointe de l’orbifold associé $ℳ:=\Gamma \setminus X$. Munissant $T^1ℳ$ de sa mesure de Patterson-Sullivan $m$, nous obtenons une estimation asymptotique de la masse d’un petit voisinage horocyclique de $𝒫$, moyennant une hypothèse sur la croissance du sous-groupe parabolique associé à $𝒫$, hypothèse qui est réalisée si $X$ est symétrique de rang $1$. Nous en déduisons une estimation asymptotique du temps de retour du flot géodésique...

In the theory of rational maps an important role is played by matings. These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of double limits, that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and...

Let $G:=\mathrm{SO}{(n,1)}^{\circ }$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{ℬ}_T\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {ℬ}_T=ℳ\left({ℬ}_T\right)+O\left(ℳ{\left({ℬ}_T\right)}^{1-\eta }\right)$ for an explicit measure $ℳ$ on $H\setminus G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic settings....

During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_w^n:{\lambda }_p\left(A\right)\to {\lambda }_p\left(A\right)$ defined on the Köthe sequence space ${\lambda }_p\left(A\right)$ exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$ and any $n\in \mathbb{N}$ is obtained. Under this assumption, the principal measure of $B_w^n$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$.

We prove that an invertible zero-dimensional dynamical system has an invariant measure of maximal entropy if and only if it is an extension of an asymptotically h-expansive system of equal topological entropy.

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that in degree one this map is injective and has torsion free cokernel. We show by example, however, that, in degree one, the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.

This paper is a continuation of [1], where a explicit description of the scrambled sets of weakly unimodal functions of type 2∞ was given. Its aim is to show that, for an appropriate non-trivial subset of the above family of functions, this description can be made in a much more effective and informative way.

We consider limit cycles of a class of polynomial differential systems of the form $$\left\{\begin{array}{c}\dot{x}=y,\hfill \\ \dot{y}=-x-\epsilon (g_{21}\left(x\right)y^{2\alpha +1}+f_{21}\left(x\right)y^{2\beta })-{\epsilon }^2(g_{22}\left(x\right)y^{2\alpha +1}+f_{22}\left(x\right)y^{2\beta }),\hfill \end{array}\right.$$ where $\beta$ and $\alpha$ are positive integers, $g_{2j}$ and $f_{2j}$ have degree $m$ and $n$, respectively, for each $j=1,2$, and $\epsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$ using the averaging theory of first and second order.