A modification of the method of geometric models is proposed and applied to the study of multiplicity functions of group extensions.

It is proved that, for some generic set of the automorphisms T of the Lebesgue space with respect to the standard topology, for any M\subseteq {\mathbb N} \cup \{\infty\}(1\in M) there exists a generic set of weakly mixing group extensions T' of transformation T with M(T')=M, where M(T) denotes the set of essential spectral multiplicities of the unitary operator corresponding to the transformation T.

Dans cet article nous allons commencer par démontrer un résultat perturbatif, nous en déduirons différentes manières de créer des connexions à l'aide de perturbations de difféomorphismes petites en topologie C1. Ensuite, nous nous intéresserons plus spécifiquement à la place des variétés stables des points périodiques hyperboliques ainsi qu'à la fermeture d'orbites récurrentes en un sens faible; puis, nous ferons une étude plus fine des difféomorphismes de classe C1 ‘génériques’, montrant en particulier l'existence d'attracteurs, au sens de Liapounov, transitif en un certain sens faible ; enfin, nous expliquerons ce qui se passe pour les difféomorphismes qui préservent une forme volume finie.

We show that the n-center problem in \mathbb{R}3 has positive topological entropy for n\ge 3. The proof is based on global regularization of singularities and the results of Gromov and Paternain on the topological entropy of geodesic flows. The n-center problem in S3 is also studied.

For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non-uniformly hyperbolic dynamical systems developed by L. S. Young.

Let ({\cal T}^1 S, m, (T^t){t \in \mathbb{R}}) be the geodesic flow on the unit tangent bundle of a surface S of negative constant curvature and finite volume. We show that every Hölder function on {\cal T}^1 S is, for the discrete time action of the geodesic flow, homologous to a martingale increment. From this representation follow the central limit theorem and its improvements, and a characterization of Hölder functions which are coboundaries in the class of measurable functions.

Soit ({\cal T}^1 S, m, (T^t)_{t \in \mathbb{R}}) le flot géodésique sur le fibré unitaire d'une surface S de courbure négative constante de volume fini. Nous montrons que toute fonction höldérienne sur {\cal T}^1 S est, pour l'action du flot géodésique à temps discret, homologue à un accroissement de martingale. Cette représentation permet d'obtenir le théorème de la limite centrale et ses extensions, et de caractériser les fonctions höldériennes qui sont des cobords dans la classe des fonctions mesurables.

We find asymptotics for the number of cylinders and saddle connections on flat surfaces. These results extend previous results of Veech.

Let D=(\omega_n)_{n\ge0} be the multiplicative semi-group generated by the coprime integers q_1,\dotsc, q_\tau arranged in increasing order. If f is a real-valued 1-periodic function, we consider the sums S_nf(t)=\sum_{0\le k<n} f(\omega_kt). For a large class of functions, we prove the existence of a limiting variance \sigma^2 for the sequence \{S_nf/\sqrt n\}, we give a function characterization for the case when \sigma=0 and finally we prove a central limit theorem.

We study the convergence of N^{-1} \sum f_1(T^{a_1n}x)f_2(T^{a_2n}x)f_3(T^{a_3n}x), for a measure-preserving system (X, \mathcal{B}, \mu, T) and f_{1}, f_{2}, f_{3} \in L^{\infty}(\mu). This generalizes the theorem of Conze and Lesigne on such expressions and simplifies the proof. We also obtain a description of the limit.

For a continuous circle map T, define the barycentre of any T-invariant probability measure \mu to be b(\mu)=\int_{S^1} z\, d\mu(z). The set \Omega of all such barycentres is a compact convex subset of \mathbb{C}. If T is conjugate to a rational rotation via a Möbius map, we prove \Omega is a disc. For every piecewise-onto expanding map we prove that the barycentre set has non-empty interior. In this case, each interior point is the barycentre of many invariant measures, but we prove that amongst these there is a unique one which maximizes entropy, and that this measure belongs to a distinguished two-parameter family of equilibrium states. This family induces a real-analytic radial foliation of int(\Omega), centred around the barycentre of the global measure of maximal entropy, where each ray is the barycentre locus of some one-parameter section of the family. We explicitly compute these rays for two examples. While developing this framework we also answer a conjecture of Z. Coelho [6] regarding limits of sequences of equilibrium states.

We give an elementary proof of existence and uniqueness of Gibbs states for Hölder weight systems on subshifts of finite type. This uses a notion of duality for such subshifts. The approach of Paterson [2] is used to construct a measure with a prescribed Jacobian and the duality is used to produce an invariant measure from this

In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to zero or an upper estimate of the Hausdorff dimension of the Julia set.

Relative periodic solutions are ubiquitous in dynamical systems with continuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite-dimensional problem. Independently, Lamb and Melbourne showed how to systematically study local bifurcation from isolated periodic solutions with discrete spatiotemporal symmetries.

In this paper, we show how the center bundle theorem, when combined with certain group theoretic results, reduces bifurcation from relative periodic solutions to bifurcation from isolated periodic solutions. In this way, we obtain a systematic approach to the study of local bifurcation from relative periodic solutions.

In our paper [1], in the proof of Proposition 1, we implicitly assume that the polynomial P is monic, although later we apply this proposition to polynomials which are not monic. The following corrections should be made in the proof of Proposition 1.