We show that the entropy of a finitely generated pseudogroup (respectively, of a foliation on a compact Riemannian manifold) can be calculated by suitable counting separated pseudo-orbits (respectively, pseudoleaves).

Consider a Hamiltonian system with $d$ degrees of freedom whose motions are all linear on tori of some fixed dimension $n\led$ is such a system necessarily completely (or else non-commutatively) integrable? We show that the answer is affirmative under quite broad conditions, but not always, and we provide counterexamples

We describe the dynamical behaviour of the entire transcendental non-critically finite function $f_\lambda (z) = \lambda(e^z - 1)/z$, $\lambda > 0$. Our main result is to obtain a computationally useful characterization of the Julia set of $f_\lambda (z)$ as the closure of the set of points with orbits escaping to infinity under iteration, which in turn is applied to the generation of the pictures of the Julia set of $f_\lambda (z)$. Such a characterization was hitherto known only for critically finite entire transcendental functions [11]. We find that bifurcation in the dynamics of $f_\lambda (z)$ occurs at $\lambda = \lambda^{*}$ ($\approx 0.64761$) where $\lambda^\ast = {(x^{*})}^{2} /({e}^{x^{*}} -1)$ and $x^{*}$ is the unique positive real root of the equation $e^{x}(2 -x ) -2 = 0$.

The structure of the cohomology equivalence classes for non-singular, not necessarily invertible mappings of probability spaces is studied. In particular, some results of Kochergin and Ornstein–Smorodinsky on the structure of these classes for measure-preserving automorphisms are generalized to the case of non-singular endomorphisms. Our approach is based on Hopf's maximal ergodic theorem and its proof by Garsia.

We show that for a large class of piecewise monotonic transformations on a totally ordered, compact set one can construct conformal measures and obtain the exponential mixing rate for the associated equilibrium state. The method is based on the study of the Perron–Frobenius operator. The conformal measure, the density of the invariant measure and the rate of mixing are deduced by using an appropriate Hilbert metric, without any compactness arguments, even in the case of a countable to one transformation.

A new approach to the study of the rate of mixing in Anosov flows, recently proposed by Chernov, is simplified and generalized to the higher-dimensional case.

We show here that some finite measure on $[-\pi,\pi]$ can never be obtained as a spectral measure of a transformation induced by a rotation. For this, we propose a new way to build a Kronecker set, which leads to a non loosely Bernoulli Gaussian–Kronecker automorphism.

On montre ici qu'une certaine mesure finie sur $[-\pi,\pi]$ ne peut jamais être obtenue comme mesure spectrale d'une transformation induite par une rotation. On propose pour cela une nouvelle façon de construire un ensemble de Kronecker, qui permet de voir que certains systèmes dynamiques gaussiens–Kronecker ne sont pas lâchement Bernoulli.

Recently, Naito considered quasi-periodic orbits of Hölder continuous functions and obtained results relating the exponent in the Hölder condition to the asymptotic behaviour of the inclusion lengths of the $\epsilon$-almost periods of these orbits, and also to the fractal dimension of these orbits. In this paper we improve these results.

We prove that certain topologically mixing two-dimensional shifts of finite type have a ‘fundamental’ $1$-cocycle with the property that every continuous $1$-cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocycle. These fundamental cocycles are closely connected with representations of the shift space by Wang tilings and the tiling groups of Conway, Lagarias and Thurston, and they determine the projective fundamental groups of the shift spaces introduced by Geller and Propp.

We characterize the set of $n$-jets admitting an extension which is a germ of a differential equation with an analytic first integral, and compute its codimension in the $n$-jet space. Some applications in the case of the centre-focus problem are given.

We prove that the time-one map of the geodesic flow for a closed, negatively curved surface is stably ergodic.

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.