Not every Cantor set can arise as the minimal set for a C^1 diffeomorphism of the circle. For example, McDuff has shown that the usual ‘middle thirds’ Cantor set cannot arise in this way. In this paper we exclude a suitable neighborhood of the class of affine Cantor sets.

Let M={\tilde M}/\Gamma be a closed negatively curved manifold with universal covering {\tilde M} and fundamental group \Gamma. Every Gibbs equilibrium state \nu of a Hölder continuous function on the unit tangent bundle T^1M of M projects to a \Gamma-invariant ergodic measure class mc(\nu_+) on the ideal boundary \partial{\tilde M} of {\tilde M}. We show that this measure class is also ergodic under the action of any normal subgroup \Gamma^\prime of \Gamma for which the factor group \Gamma/\Gamma^\prime is nilpotent.

We investigate uniform ergodic type theorems for almost additive and subadditive functions on a subshift over a finite alphabet. We show that every uniquely ergodic subshift admits a uniform ergodic theorem for Banach-space-valued almost additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides, in particular, a sufficient condition for unique ergodicity.

We describe the construction of almost-periodic solutions for a particular Schrödinger equation on a finite x- interval, depending on some potential V. The construction proceeds by iterating a KAM theorem about the existence of quasi-periodic solutions for some partial differential equations. As a result, we find for ‘almost all’ potentials V uncountably many almost-periodic solutions accumulating at the zero solution.

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.

We present a certain version of the ‘non-stationary’ normal forms theory for extensions of topological dynamical systems by smooth contractions of \mathbb{R}^n. The central concept is a notion of a sub-resonance relation which is an appropriate generalization of the notion of resonance between eigenvalues. Finitely differentiable, infinitely differentiable and analytic maps are considered.

Let \Omega be a metric space. There is a natural metric that can be put on the space of all measures on \Omega called the Wasserstein metric. It is obtained by taking the infimum over all couplings of the two measures of the average distance between two points. There is a simple coupling called the greedy coupling for the two measures, but operations researchers will assure you that it is terrible to use the value obtained by the greedy coupling as a substitute for the Wasserstein distance between two measures. The purpose of this paper, however, is to show that if all that you are interested in is the topology obtained by the metric, the value you get from the greedy coupling is good enough.

The question addressed in this paper is the worst-case growth rate, for ergodic processes, in the number of conditional measures on n-steps in the future, given the past, that are a fixed distance apart. It is shown that if error is measured using the variational (i.e. distributional) distance then doubly exponential growth is possible, while if error is measured using the \bar{d}-metric then more than exponential growth is possible. The question of whether doubly exponential growth is possible in the \bar{d}case is left open.

We compute the exact lower bounds for some averages arising in ergodic theory. In particular, we prove that for any measure-preserving system (X,\mathcal{B},\mu,T) with \mu(X)<\infty, any A\in\mathcal{B} and any N\in\mathbb{N}, N^{-1}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}A)\geq\sqrt{\mu(A)^{2}+(\mu(X)-\mu(A))^{2}}+\mu(A)-\mu(X).

We show the stability of an elliptic fixed point of an analytic area-preserving mapping under the Bruno condition, using Moser's Twist-mapping Theorem. We require that there are non-vanishing Birkhoff constants and prove the convergence of the transformation to the Birkhoff normal form if all Birkhoff constants are zero. The convergence theorem is formulated and proved for arbitrary dimensions.

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 prove a relative version of Krieger's Theorem constructing relative generating partitions with cardinality estimated via the relative entropy. We introduce relative asymptotically entropy expansive random transformations and construct for them relative topological generators with cardinality estimated via the relative topological entropy, which enables us to represent such systems by means of random symbolic subshifts.

In this paper, we consider random dynamical systems (abbreviated as RDSs) generated by compositions of one-sided stationary random endomorphisms of class C^2 of a compact manifold. We will first introduce the notions of entropy and Lyapunov exponents for such RDSs, and then prove that the entropy formula of Pesin type holds if the sample measures of an invariant measure are absolutely continuous with respect to the Lebesgue measure on the manifold. Our result covers those obtained by Ledrappier and Young [11], and Liu [12] for i.i.d. random diffeomorphisms or (non-invertible) endomorphisms, and that obtained by [12] for two-sided stationary random endomorphisms. If the phase spaces are compact and finite-dimensional manifolds without boundary, this result may be considered as the almost-final form of Pesin entropy formula for RDSs with absolutely continuous invariant or sample measures.

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

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dots, over a finite alphabet A, we define the maximal pattern complexity by p_\alpha^*(k)=\sup_\tau\sharp\{\alpha_{n+\tau(0)} \alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)}; n=0,1,2,\dots\} where the ‘sup’ is taken over all subsequences 0=\tau(0)<\tau(1)<\dots<\tau(k-1) of integers of length k. We prove that \alpha is eventually periodic if and only if p_\alpha^*(k)\le 2k-1 for some k. Infinite words \alpha, with p_\alpha^*(k)=2k for any k, are called pattern Sturmian words and are studied.

The connected configuration space of a so-called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with specular reflections at the boundaries of the removed cylinders. It is proven here that under a certain geometric condition a cylindric billiard flow is completely hyperbolic. As a consequence, every hard ball system is completely hyperbolic.

Any Axiom A holomorphic map on \mathbb{P}^2\mathbb{C} has a natural ergodic measure \mu. The set K^- is the set of points which have arbitrarily close neighbors with prehistories not convergent to supp \mu.

This set is the analogue of the set of points with bounded backwards iterates from the case of Hénon mappings. For Hénon diffeomorphisms it was shown by Bedford and Smillie that K^- either has an empty interior or \mathop{K}\limits^\circ{}^{-}= union of the basins of finitely many repelling periodic points. For s-hyperbolic holomorphic endomorphisms of \mathbb{P}^2, we show here that the only possibility is \mathop{K}\limits^\circ{}^{-} = \emptyset. This answers a question of Fornaess.

We also prove that \mathop{K}\limits^\circ{}^{-} is the union of finitely many repelling basins when the endomorphism is a perturbation of a Hénon mapping.

Several non-trivial examples of s-hyperbolic maps are discussed as well, some of them coming from perturbations of product maps and others from solenoids.

An integrable discrete-time Lagrangian system on the group of area-preserving plane diffeomorphisms \mathit{SDiff}(\mathbb{R}^{2}) is considered. It is shown that non-trivial dynamics exists only for special initial data and the corresponding mapping can be interpreted as a Bäcklund transformation for the (simple) Monge–Ampère equation. In the continuous limit, this gives the isobaric (constant pressure) solutions of the Euler equations for an ideal fluid in two dimensions. In the Appendix written by E. V. Ferapontov and A. P. Veselov, it is shown how the discrete system can be linearized using the transformation of the simple Monge–Ampère equation going back to Goursat.

An extension of the Shannon–McMillan–Breiman Theorem to a class of non-commutative dynamical systems is given.

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dotsc over a finite alphabet, the authors introduced a new notion of complexity called maximal pattern complexity defined by p_\alpha^*(k):=\sup_\tau\sharp\{\alpha_{n+\tau(0)}\alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)};n=0,1,2,\dotsc\} where the supremum is taken over all sequences of integers 0=\tau(0)<\tau(1)<\dotsc<\tau(k-1) of length k. The authors proved that \alpha is aperiodic if and only if p_\alpha^*(k)\ge 2k for every k=1,2,\dotsc. A word \alpha with p_\alpha^*(k)=2k for every k\geq 1 is called pattern Sturmian. In this paper, we give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations. We also investigate the maximal pattern complexity of various discrete dynamical systems including irrational rotations on the circle, and self-similar systems generated by substitutions. We show that, for each irrational rotation on the circle, there exists a twofold partition of the circle, with respect to which the system generated has full maximal pattern complexity with probability one. Using the arithmetic properties of the underlying numeration system associated to a substitution dynamical system, we prove that the maximal pattern complexity of the fixed point of the Rauzy substitution 1\mapsto 12, 2\mapsto 13, 3\mapsto 1 has exponential growth. It is well known that the system generated by the Rauzy substitution is isomorphic in measure to an irrational rotation on the 2-torus.