For a fixed C^*-algebra A, we consider all non-commutative dynamical systems that can be generated by A. More precisely, an A-dynamical system is a triple (i,B,\alpha) where \alpha is a *-endomorphism of a C^*-algebra B, and i: A\subseteq B is the inclusion of A as a C^*-subalgebra with the property that B is generated by A\cup\alpha(A)\cup\alpha^2(A)\cup\dotsb. There is a natural hierarchy in the class of A-dynamical systems, and there is a universal one that dominates all others, denoted (i,\mathcal{P}A,\alpha). We establish certain properties of (i,\mathcal{P}A,\alpha) and give applications to some concrete issues of non-commutative dynamics.

For example, we show that every contractive completely positive linear map \varphi:A\to A gives rise to a unique A-dynamical system (i,B,\alpha) that is ‘minimal’ with respect to \varphi, and we show that its C^*-algebra B can be embedded in the multiplier algebra of A\otimes\mathcal{K}.

For diffeomorphisms on 3-manifolds, two-dimensional links composed of tori (and possibly one Klein bottle) embedded in S^2\times S^1 appear in a natural way to describe the topological position of two-dimensional unstable manifolds of saddles in the basin of a periodic sink. For such a link \mathcal{E} we build a diffeomorphism f_{\mathcal{E}} which is used as a canonical model for ‘pieces of the dynamics’, namely the basin of attractors consisting of a sink and a finite set of two-dimensional unstable manifolds of saddles.

In this paper, which is a sequel to an earlier paper, we consider special examples, methods, and invariants for the isomorphism problem for ordered dimension groups built on square primitive integer matrices A. We use the algorithm of our first paper to derive techniques and computable invariants for deciding isomorphism for subclasses of the matrices A.

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.

Let G be a discrete group which admits an amenable action on a compact space and \gamma \in \textrm{Aut}(G) be an automorphism. We define a notion of entropy for \gamma and denote the invariant by ha(\gamma). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(\gamma) = h_{\rm Top}(\hat{\gamma}) where h_{\rm Top}(\hat{\gamma}) denotes the topological entropy of the induced automorphism \hat{\gamma} of the (compact, abelian) dual group \hat{G}.

ha(\cdot) enjoys a number of basic properties which are useful for calculations. For example, it decreases in invariant subgroups and certain quotients. These basic properties are used to compute the dual entropy of an arbitrary automorphism of a crystallographic group.

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.

Analytic quasi-periodic skew-systems with close to constant coefficients and with Diophantine frequencies seem to be, in a measure theoretical sense, ‘generically’ reducible. In this paper we show that such systems are, in a topological sense, generically uniquely ergodic and, in particular, non-reducible.

We prove the existence of a Sinai–Ruelle–Bowen (SRB) measure and the exponential decay of correlations for smooth observables for mixing Anosov C^{1+\alpha} diffeormorphisms on a d-dimensional (d \geq 2) Riemannian manifold. The novelty lies in the very simple method of proof. We construct explicitly a coupling between two initial densities so that under the action of the diffeomorphism, both components get closer and closer. The speed of this convergence can be explicitly estimated and is directly related to the speed of decay of the correlations. The existence of the SRB measure and its properties readily follow.

We answer in the negative a question of Viviane Baladi: in contrast to its deterministic or averaged versions, the random zeta function has no meromorphic extensions, hence no poles which would describe the Lyapunov exponents of the corresponding random transfer operator. In fact we show that the sequence of traces of the random iterates of this operator does not determine almost surely these exponents.

We show that, for any compact surface, there is a residual (dense G_{\delta}) set of C^{1} area-preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was announced by R. Mañé, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous SL(2,\mathbb{R})-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

Dans cet article, nous complétons la classification analytique des feuilletages analytiques réduits de codimension 1 en dimension n\ge3, dans les cas résonnants. Nous montrons que, génériquement, l'espace de modules se réduit à un point.

Let G be a connected Lie group and \mathcal{C}(G) be the space of all closed subgroups of G, equipped with the Fell topology. We consider actions on \mathcal{C}(G), of G (by conjugation) and of groups of Lie automorphisms of G (by set operation). We describe conditions under which the orbit of a closed subgroup with finitely many connected components is locally closed (open in its closure) in \mathcal{C}(G). The results are applied to conclude, in particular, that for any ergodic action of G on a standard Borel space the connected components of the identity in almost all the stabilizers are isomorphic to each other, and that if all the stabilizers are assumed to have only finitely many connected components then almost all of them are isomorphic to each other. Various conditions are also described under which almost all the stabilizers are conjugate to each other.

In singular perturbation problems one frequently has a family (\pi_\eps)_{\eps\in]0,1]} of semiflows on a space X and a ‘singular limit flow’ \pi_0 of this family which is defined only on a subspace X_0 of X. Moreover, \pi_\epsilon converges to \pi_0 only in some ‘singular’ sense. Such a situation occurs, for example, in fast–slow systems of differential equations or in evolution equations on thin spatial domains.

In this paper we prove a general singular Conley index continuation principle stating that every isolated invariant set K_0 of \pi_0 can be continued to a nearby family K_\epsilon of isolated invariant sets of \pi_\epsilon with the same Conley index. We illustrate this continuation result with damped wave equations on thin domains. This extends some results from our previous work.

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.

\widetilde{J}-holomorphic cylinders in \mathbb{R}\times M, where M is a contact manifold are investigated. It is shown that a sufficiently long cylinder having small area is close to a constant map, if its center action vanishes. If its center action is positive, it is close to a cylinder over a periodic orbit of the Reeb vector field, and has a well-determined shape. A compactness result is deduced, which is useful in symplectic field theory.

Soit (\Omega,{\cal B},m) un espace de probabilité. À une action T stationnaire de \mathbb{Z}^d est associée une notion algébrique de cocycle de degré \geq 1, le degré 1 correspondant au sens habituel. Celle-ci a été étudiée entre autres par Westman (1971), Feldman et Moore (1977). Une autre notion de cocycle, présentant des analogies avec le calcul différentiel, à été étudiée par A. et S. Katok (1995). Nous présentons un mécanisme de génération associant, à tout cocycle de ce dernier type, un cocycle algébrique de même degré. De plus, toutes les classes de cohomologie algébrique peuvent être engendrées ainsi. Ce résultat a pour conséquence que tout cocycle algébrique de degré \geq d+1 est un cobord. Il permet aussi de montrer que dans la classe des cocycles intégrables, la cohomologie algébrique de degré \leq d n'est pas triviale, alors qu'elle l'est dans la classe des cocycles mesurables dès le degré 2. Enfin, la méthode de génération présentée permet d'obtenir des cocycles algébriques vérifiant les théorèmes ergodiques correspondant à leur degré.

Relative transfer operators are used in Denker and Gordin's 1999 paper to construct Gibbs families of conditional measures on fibres of a uniformly-expanding and exact-fibred system. Here we investigate the relationship between these operators and the relative variational principle (extending results for non-fibred expanding systems).

It turns out that the maximal value for the free energy in the relative variational problem can be represented in terms of the transfer operator. However, for a general potential, the possibility to reduce the construction of an equilibrium measure to the search for an appropriate family of conditional measures on the fibres, critically depends on the invertibility of the base transformation.

A certain class of potentials (called basic) which allow the last-mentioned reduction is introduced and the properties of the corresponding equilibrium measures are studied. Any measure of this kind gives rise to a regular factor; under a natural assumption the latter property is shown to be equivalent to the validity of the relative version of Rokhlin's formula for the entropy of a measure preserving transformation. Several examples are presented, among them families of polynomial skew products in \mathbb{C}^n restricted to their Julia sets.

Using the orbital approach to the entropy theory we extend from \mathbb{Z}-actions to general countable amenable group actions T (or provide new short proofs for) the following results: (1) the relative and absolute Krieger Theorem about finite generating partitions (and its infinite Rokhlin counterpart in the case h(T)=\infty); (2) the relative and absolute Sinai Theorem about Bernoullian factors; (3) the Thouvenot Theorem that every intermediate factor of a relatively Bernoullian action is also relatively Bernoullian; (4) the Thouvenot Theorem that a factor of T with the strong Pinsker property enjoys this property; (5) the Smorodinsky–Thouvenot Theorem that T can be spanned by three Bernoullian factors; (6) the Ornstein–Weiss isomorphism theory for Bernoullian actions of the same entropy (provided that they possess generating partitions with at least three elements); and (7) there are uncountably many non-equivalent completely positive entropy extensions of T of the same relative entropy.

In proving these theorems, we were able to bypass the machinery of Orstein and Weiss except for the Rokhlin lemma. It is shown that the language of orbit equivalence relations and their cocycles (unlike the standard dynamical one) is well suited for the inducing operation needed to settle (1), (5) and (6).

In this paper, we continue our investigation of dissipative PDE's forced by random bounded kick-forces and of the corresponding random dynamical system (RDS) in function spaces. It has been proved that the domain \mathcal{A} of attainability from zero (which is a compact subset of a function space) is invariant for the RDS associated with the original equation and carries a stationary measure \mu, which is unique among all measures supported by \mathcal{A}. Here we show that \mu is the unique stationary measure for the RDS in the whole space and study its ergodic properties.

We introduce transversality conditions for a piecewise smooth map in any dimension with a sufficiently regular discontinuity set. We prove that maps satisfying these conditions have certain nice properties and, among all maps with a common set of discontinuities, form an open and dense subset. This, in conjunction with other works of the author, implies the existence of stochastically stable invariant densities for an open and dense set of piecewise smooth expanding maps.