We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Hölder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters.

This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.

We show that a periodic orbit of large period of a diffeomorphism or flow either admits a dominated splitting of a prescribed strength or can be turned into a sink or a source by a $C^1$-small perturbation along the orbit. As a consequence we show that the linear Poincaré flow of a $C^1$-vector field admits a dominated splitting over any robustly transitive set.

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is a special case of the well known concept of an inertial manifold for more general systems. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. The aim of this paper is to extend this result in two directions: firstly, where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney extension theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.

We consider $SL(2, {\Bbb R} )$-valued cocycles over rotations of the circle and prove that they are likely to have Lyapunov exponents $\approx \pm \log \lambda $ if the norms of all of the matrices are $\approx \lambda $. This is proved for $\lambda $ sufficiently large. The ubiquity of elliptic behavior is also observed.

A class of locally isometric, but not necessarily invertible mappings of an interval is considered. We show that under some conditions the study of the dynamical properties of these mappings can be reduced to interval exchange transformations. On the other hand, there are examples of mappings in this class with ergodic invariant measures supported by Cantor sets. The so-called μβ -sets studied by Y. Katznelson appear naturally in such examples.

We consider an Abel equation $(*)$ $y^{\prime}=p(x)y^2+q(x)y^3$ with $p(x)$, $q(x)$ polynomials in $x$. A center condition for ($*$) (closely related to the classical center condition for polynomial vector fields on the plane) is that $y_0=y(0)\equiv y(1)$ for any solution $y(x)$ of ($*$). This condition is given by the vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. A new basis for the ideals $I_k=\{v_2,\dots,v_k\}$ has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of $P=\int p$ and $Q=\int q$ in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments $\int P^kq$. On this base, explicit center equations are obtained for small degrees of $p$ and $q$.

Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to $C^* (E/G)$. Since every connected graph $E$ may be expressed as the quotient of a tree $T$ by an action of a free group $G$ we may use our results to show that $C^* (E)$ is strongly Morita equivalent to the crossed product $C_0 ( \partial T ) \times G$, where $\partial T$ is a certain zero-dimensional space canonically associated to the tree.

We prove that there is a residual subset $\mathcal{I}$ of ${\rm Diff}^1({\it M})$ such that any homoclinic class of a diffeomorphism $f\in \mathcal{I}$ having saddles of indices $\alpha$ and $\beta$ contains a dense subset of saddles of index $\tau$ for every $\tau\in [\alpha,\beta]\cap \mathbb{N}$. We also derive some consequences from this result about the Lyapunov exponents of periodic points and the sort of bifurcations inside homoclinic classes of $C^1$-generic diffeomorphisms.

We describe a family of renormalization group transformations which apply to the problem of stability of invariant tori for analytic Hamiltonians. Each of these transformations corresponds to a fixed rotation vector, within a certain class of vectors in ${\mathbb R}^d$. As an application, we prove a KAM-type theorem that yields analytic invariant $d$-tori depending analytically on the Hamiltonian.

Let φ be an Axiom A flow restricted to a basic set, let g be a C∞ function and let , where λg(τ) is the g length of the closest orbit τ, λ(τ) is the period of τ and h is the topological entropy of φ. We obtain an asymptotic formula for πg which includes the ‘prime number’ theorem for closed orbits. This result generalizes Bowen's theorem on the equidistribution of closed orbits. After establishing an analytic extension result for certain zeta functions the proofs proceed by orthodox number theoretical techniques.

We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.

We show that piecewise isometries, i.e. non-necessarily invertible maps defined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope, have zero topological entropy in any dimension. This had been conjectured by a number of authors. The proof is by an induction on the dimension and uses a device (the differential of a piecewise linear map) introduced by M. Tsujii.

We apply Thurston's equivalence theory between dynamical systems of post-critically finite branched coverings and rational maps, to try to construct, from a pair of polynomials, a rational map. We prove that given two post-critically finite quadratic polynomials fc: z→z2+c and fc:z→ z2+c′, one can get a rational map if and only if c, c′ are not in conjugate limbs of the Mandelbrot set.

Deng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2λu > min{−λs, λuu} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.

We study the discrete quasi-periodic Schrödinger equation \[-(u_{n+1}+u_{n-1})+\lambda V(\theta+n\omega)u_n=Eu_n\] with a non-constant C1 potential function $V:\mathbb{T}\to\mathbb{R}$. We prove that for sufficiently large $\lambda$ there is a set $\Omega\subset\mathbb{T}$ of frequencies $\omega$, whose measure tends to 1 as $\lambda\to\infty$, with the following property. For each $\omega\in\Omega$ there is a ‘large’ (in measure) set of energies E, all lying in the spectrum of the associated Schrödinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrödinger cocycle is minimal but not ergodic.

A C∞ metric is constructed on S2 whose geodesic flow has positive measure entropy.

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to $\unicode[STIX]{x1D707}$-mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

Parmi les systèmes dynamiques définis par des substitutions, les mieux connus sont ceux qui proviennent de substitutions à longueur constante. F. M. Dekking [1] a déterminé les valeurs propres de ces systèmes, et, plus récemment, M. Queffelec a étudié leur type spectral maximal. On se propose ici de déterminer les valeurs propres de ces systèmes dans le cas général des substitutions à longueur variable; dans cette direction, un résultat partiel a été obtenu par J. C. Martin [3] dans le cas des substitutions de 2 lettres, pour les valeurs propres correspondant à des fonctions propres continues.

We generalize several results of the classical theory of thermodynamic formalism by considering a compact metric space $M$ as the state space. We analyze the shift acting on $M^{\mathbb{N}}$ and consider a general a priori probability for defining the transfer (Ruelle) operator. We study potentials $A$ which can depend on the infinite set of coordinates in $M^{\mathbb{N}}$. We define entropy and by its very nature it is always a non-positive number. The concepts of entropy and transfer operator are linked. If $M$ is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case $M=S^{1}$, and the a priori measure is Lebesgue $dx$, the infinite product of $dx$ on $(S^{1})^{\mathbb{N}}$ will have zero entropy. We analyze the Pressure problem for a Hölder potential $A$ and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the thermodynamic formalism for the Bernoulli space with countable infinite symbols. Moreover, the so-called $XY$ model also fits under our setting. In this last case $M$ is the unitary circle $S^{1}$. We explore the differentiable structure of $(S^{1})^{\mathbb{N}}$ by considering a certain class of smooth potentials and we show some properties of the corresponding main eigenfunctions.