A selection of basic results in smooth ergodic theory and in the thermodynamic formalism of dynamical systems generated by compositions of random maps is reviewed in this article.

We discuss the dynamics of a class of non-ergodic piecewise affine maps of the torus. These maps exhibit highly complex and little understood behavior. We present computer graphics of some examples and analyses of some with a decreasing degree of completeness. For the best understood example, we show that the torus splits into three invariant sets on which the dynamics are quite different. These are: the orbit of the discontinuity set, the complement of this set in its closure, and the complement of the closure. There are still some unsolved problems concerning the orbit of the discontinuity set. However we do know that there are intervals of periodic orbits and at least one infinite orbit. The map on the second invariant set is minimal and uniquely ergodic. The third invariant set is one of full Lebesgue measure and consists of a countable number of open octagons whose points are periodic. Their orbits can be described in terms of a symbolism obtained from an equal length substitution rule or the triadic odometer.

In this paper we study the dynamics of a rational function \phi\in K(z) defined over some finite extension K of \mathbb{Q}_p. After proving some basic results, we define a notion of ‘components’ of the Fatou set, analogous to the topological components of a complex Fatou set. We define hyperbolic p-adic maps and, in our main theorem, characterize hyperbolicity by the location of the critical set. We use this theorem and our notion of components to state and prove an analogue of Sullivan's No Wandering Domains Theorem for hyperbolic maps.

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.

It is shown that regional proximality is an equivalence relation in a minimal flow which contains a closed proximal cell. Also, a certain group theoretic condition is shown to be equivalent to a ‘local Bronstein’ condition.

Let \Gamma be a non-solvable pseudogroup of holomorphic transformations in one variable fixing zero. Then for any z_0 sufficiently near zero and outside some real analytic set containing zero and depending on \Gamma, for any germ of biholomorphism \phi defined at z_0 with \phi(z_0)-z_0 sufficiently small, there exists a sequence \gamma_n \in \Gamma which tends to \phi uniformly on some neighborhood of zero. If we let \Gamma be the holonomy pseudogroup of a compact leaf, we find that holomorphic codimension-1 foliations with non-solvable holonomy admit no transverse geometric structure in addition to the conformal one. This applies, in particular, to the singular foliation induced on \mathbb{C}\mathbb{P}^2 by the differential equation dw/dz = P_n(z,w)/Q_n(z,w) where P_n and Q_n are the generic polynomials of degree n, hence the title of this paper.

The theory of Fatou and Julia is extended to include the dynamics of functions f which are meromorphic in \widehat{\mathbb{C}} outside a totally disconnected compact set E(f) at whose points the cluster set of f is \widehat{\mathbb{C}}. The Julia set is defined not only by the standard approach but is also characterized in terms of the set of points whose orbits approach a point of E(f). For the subclass where E(f) has a complement of class OAD and the inverse of f has a finite set of singular points it is shown that neither wandering components nor Baker domains occur in F(f)>. As an application, functions of a certain general class are shown to have a totally disconnected Julia set.

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.

Let BV(I) be the (Banach) space of real functions h on I:=[0,1] of bounded variation \mathop{\rm var}\nolimits h with the norm \Vert h\Vert = \vert h\vert + \mathop{\rm var}\nolimits h, where \vert h\vert :=\sup_{t\in I}\vert h(t)\vert. We introduce a condition (M) on the monotony of the transition probability function (TPF) of the piecewise monotonic transformation \tau associated with an f-expansion. The transfer operator associated with \tau is denoted by U. Under (M) we can apply methods used in the continued fraction expansion case to U acting on BV(I). An expression in terms of the TPF of a contraction constant, \theta, of \mathop{\rm var}\nolimits Uh is thus obtained which is subsequently used to state a Gauss–Kuzmin–Levy theorem. This result yields the answer in a special case to the problem of determining a numerical value of the constant \theta occurring in the Kuzmin-type theorem proved in [1].

Examples of numerical values of \theta in particular cases are given. Possible extensions of the methods used are mentioned.

We define a normalized entropy functional for compact Finsler manifolds of negative flag curvature. Using the method of Besson, Courtois, and Gallot, we show that among all such manifolds that are homotopy equivalent to a compact, Riemannian, locally symmetric manifold of negative curvature, the entropy functional is minimized precisely on the locally symmetric manifold.

Soient f un endomorphisme holomorphe de \mathbb{C}\mathsf{P}^k de degré au moins 2 et \mu sa mesure d'équilibre. Nous montrons que l'extension naturelle de (f,\mu) possède la propriété de Bernoulli.

We show that Chacon's non-singular type III_\lambda transformation T_\lambda, 0<\lambda\leq 1, is power weakly mixing, i.e. for all sequences of non-zero integers \{k_{1},\dotsc,k_{r}\}, T_\lambda^{k_{1}}\times\dotsb\times T_\lambda^{k_{r}} is ergodic. We then show that in infinite measure, this condition is not implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.

Let (B,T) be a fibred system, where B is a compact and convex subset of \mathbb{R}^{d} and T:B\mapsto B is a map. Under adequate norm the space \mathcal{L} (\mathcal{H}) (respectively \mathcal{C} (\mathcal{H})) of Lipschitz continuous (respectively continuous) functions on every cell H of a certain finite partition \mathcal{H} of B is a Banach space. We delimit by Conditions \rm (A),\dotsc,(E),(F)' a class of fibred systems, including multidimensional continued fractions. We first prove that the Perron–Frobenius operator A associated with T under the Lebesgue measure \lambda considered on <formula form="inline" disc="math" id="eq014"><formtex notation="amstex">\mathcal{L} (\mathcal{H}) is weak-mixing. Using the ergodic theorem of Ionescu–Tulcea and Marinescu we then prove that there exist positive constants q >1 and M for which

\vert\mspace{-2mu}\vert\mspace{-2mu}\vert A^{n}\varphi -h\int_{B}\varphi \,d\lambda\vert\mspace{-2mu}\vert\mspace{-2mu}\vert \leq q^{n}M\vert\mspace{-2mu}\vert\mspace{-2mu}\vert\varphi \vert\mspace{-2mu}\vert\mspace{-2mu}\vert, \quad n\in \mathbb{N}, \varphi \in \mathcal{L} (\mathcal{H}),

where \vert\mspace{-2mu}\vert\mspace{-2mu}\vert\cdot\vert\mspace{-2mu}\vert\mspace{-2mu}\vert is the norm in \mathcal{L} (\mathcal{H}) and h is the density of the (unique) T-invariant probability \mu on the Borel \sigma-algebra \Sigma in B. The analogous result for the Perron–Frobenius operator U of T under \mu is also proved.

Let \varphi be a primitive, non-periodic substitution. The tiling space \mathcal{T}_\varphi has a finite (non-zero) number of asymptotic composants. We describe the form and make use of these asymptotic composants to define a closely related substitution \varphi^* and prove that for primitive, non-periodic substitutions \varphi and \chi, \mathcal{T}_\varphi and \mathcal{T}_\chi are homeomorphic if and only if {\varphi^*} (or its reverse) and \chi^* are weakly equivalent. We also provide examples indicating that for substitution minimal systems, flow equivalence and orbit equivalence are independent.

The notion of isomorphism on stable AF-C^{\ast}-algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e. is associated with a single square primitive incidence matrix. A C^{\ast}-isomorphism induces an equivalence relation on these matrices, called C^{\ast}-equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e. there is an algorithm that can be used to check in a finite number of steps whether two given primitive matrices are C^{\ast}-equivalent or not. Special cases of this problem will be considered in a forthcoming paper.

We study \mathcal{C}^k-diffeomorphisms, k\ge 1, f: M\to M, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). We prove that if f can not be \mathcal{C}^k-approximated by diffeomorphisms with homoclinic tangencies, then f is in the closure of an open set \mathcal{U}\subset \operatorname{Diff}^k(M) consisting of diffeomorphisms g with a non-hyperbolic transitive set \Lambda_g which is locally maximal and strongly partially hyperbolic (the partially hyperbolic splitting at \Lambda_g has three non-trivial directions). As a consequence, in the case of 3-manifolds, we give new examples of open sets of \mathcal{C}^1-diffeomorphisms for which residually infinitely many sinks or sources coexist (\mathcal{C}^1-Newhouse's phenomenon). We also prove that the occurrence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.

We study exactness and maximal automorphic factors of C^3 unimodal maps of the interval. We show that for a large class of infinitely renormalizable maps, the maximal automorphic factor is an odometer with an ergodic non-singular measure. We give conditions under which maps with absorbing Cantor sets have an irrational rotation on a circle as a maximal automorphic factor, as well as giving exact examples of this type. We also prove that every C^3 S-unimodal map with no attractor is exact with respect to Lebesgue measure. Additional results about measurable attractors in locally compact metric spaces are given.

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.

Let \{U_{g}\}_{g\in G} be a weakly mixing unitary action of a finitely-generated nilpotent group on a Hilbert space \mathcal{H}. Extending a known result about abelian groups, we show that weakly wandering vectors are dense in \mathcal{H}.

We prove that on any surface there is a C^\infty diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoe \Lambda which is equivalent to the unique non-trivial Hausdorff measure in \Lambda. The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoe \Lambda are relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.