In this paper we prove the existence of an entire transcendental function of arbitrarily slow growth which exhibits simple dynamics on its Fatou set.

Soient f un endomorphisme holomorphe de degrée plus grand que 2, et \mu sa mesure d'équilibre. Alors (f,\mu) posséde la propriété de Bernoulli.

For any \varepsilon > 0, we construct an explicit smooth Riemannian metric on the sphere S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is \varepsilon-dense in the unit tangent bundle. Moreover, for any \varepsilon > 0, we construct a smooth Riemannian metric on S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than \varepsilon.

We study the cyclicity of period annuli (or annulus) for general degenerate quadratic Hamiltonian systems with an elliptic segment or a saddle loop, under quadratic perturbations. By using geometrical arguments and studying the respective Abelian integral based on the Picard–Fuchs equation, it is shown that the cyclicity of period annuli (or annulus) for such systems equals two. This result, together with those of Gavrilov and Iliev (2000), Iliev (1996), Zhao et al (2000) and Zhao and Zhu (2001) gives a complete solution to the infinitesimal Hilbert 16th problem in the case of degenerate quadratic Hamiltonian systems under quadratic perturbations.

If T is an interval exchange transformation we denote by C_{{\rm aff}}(T) (respectively S_{{\rm aff}}(T)) the set of piece-wise affine maps of the interval which are conjugate (respectively semi-conjugate) to T. In this work we will give a description of the set C_{{\rm aff}}(T) for almost allT. We present an explicit interval exchange T_0 such that S_{{\rm aff}}(T_0)\backslash C_{{\rm aff}}(T_0) is non-empty. All the elements of S_{{\rm aff}}(T_0)\backslash C_{{\rm aff}}(T_0) are uniquely ergodic and have a unique wandering interval.

We show that a smooth compact Riemannian manifold of dimension greater than or equal to 2 admits a Bernoulli diffeomorphism with non-zero Lyapunov exponents.

We give an example of a strictly positive analytic reparametrization (or time change) of an irrational flow on \mathbb{T}^3 that is mixing. As an immediate application we obtain perturbations of completely integrable Hamiltonian systems that display many invariant tori on which the restricted dynamics is mixing.

We consider cocycles with negative Lyapunov exponents defined over a hyperbolic dynamical system. It is well known that such systems possess invariant graphs and that under spectral assumptions these graphs have some degree of Hölder regularity. When the invariant graph has a slightly higher Hölder exponent than the a priori lower bound on an open set (even on just a set of positive measure for certain systems), we show that the graph must be Lipschitz or (in the Anosov case) as smooth as the cocycle.

We investigate the iterative dynamics of symplectic piecewise-affine elliptic rotation maps associated with matrices with non-integer entries, \big(\begin{smallmatrix} 0 & -1 \\ 1 & a \end{smallmatrix}\big), where -2 < a < 2. We explain how their singularity and periodicity structures develop in general. In a special case, a = \pm \sqrt 2, we completely determine the orbit structure, the singularity structure, the invariant fractal and the ergodicity on the invariant fractal with respect to its Hausdorff measure. We also report an example of a period tripling cascade that arises from this system.

We investigate joinings of strongly irreducible totally non-symplectic Anosov {\mathbb Z}^k, k\ge 2 actions by toral automorphisms. We show that the existence of a non-trivial joining has strong implications for these actions, in particular, that the restrictions of the actions to a finite index subgroup of {\mathbb Z}^k are conjugate over {\mathbb Q}. We also obtain a description of the joining measures modulo the classification of zero entropy measures for the actions.

A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thom's Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times); this can also be applied to some gradient vector fields.

Dans cet article, nous étudions le problème de conjugaison par homéomorphismes des échanges d'intervalles affines aux échanges d'intervalles linéaires. Nous montrons que la C^2-conjugaison (C^1 dans le cas n=3) aux échanges linéaires sans orbite périodique implique la C^\omega-conjugaison. Ceci nous permet de donner une caractérisation simple des échanges d'intervalles affines qui sont C^2-conjugués aux échanges linéaires sans orbite périodique. D'autre part, nous construisons une famille (f_t, t\in ]0,1]) continue à un paramétre d'échanges d'intervalles affines uniquement ergodiques qui sont C^0 mais jamais C^1-conjugués au linéaire f_1, sauf f_1 lui-même, et nous indiquons un exemple d'échange d'intervalles affine (non minimal) qui est C^1 mais pas C^2-conjugué aux linéaires. Ensuite, nous donnons des critéres de conjugaison C^0.

We give elementary proofs of two rigidity results. The first one asserts that the maximal entropy measure \mu_f of a rational map f is singular with respect to any given conformal measure excepted if f is a power, Tchebychev or Lattès map. This is a variation of a result of Zdunik. Our second result is an improvement of a theorem of Fisher and Urbański. It gives a sharp description of the exceptional functions that admit invariant line fields which are defined with respect to certain invariant measures

Let f:M\rightarrow M be a C^2 diffeomorphism of a compact Riemannian manifold of dimension m\geq 2 leaving invariant an ergodic Sinai–Ruelle–Bowen measure \mu with non-zero Lyapunov exponents. We prove that \mu can be approximated by ergodic measures supported on hyperbolic horseshoes with arbitrarily large unstable dimensions.

We associate to each non-singular action A of a connected Lie group G on a closed manifold M and each A-invariant probability measure \mu on M, a linear mapping P_A from the cohomology H^*(\mathcal{F}) of the foliation \mathcal{F} given by the orbits of A onto the cohomology H^*(\mathcal{G}) of the Lie algebra \mathcal{G} of G. This extends to actions the asymptotic cycles of S. Schwartzman and D. Sullivan and gives a strong connection between the ergodic properties of A and the characteristic mapping \chi of \mathcal{F} as defined by dos Santos (Contemp. Math.161 (1994), 41–57). The ergodic notion of E-action introduced by Arraut and dos Santos (Topology31 (1992), 545–555) extends naturally to actions of Lie groups and we prove the main result of this article which shows that a non-singular E-action A of a nilpotent Lie group G completely determines the characteristic mapping \chi of the foliation \mathcal{F} given by A. This shows the strength of the above connection and gives as a corollary the vanishing theorem for foliated bundles proved by dos Santos.

Let M be a three-dimensional compact connected oriented manifold and f : M \rightarrow M an expansive diffeomorphism. We prove that non-wandering points have local stable or unstable sets locally separating M. This property allows us to prove that if \Omega(f)=M then f is conjugate to a linear Anosov diffeomorphism and M=T^3, the three-dimensional torus.