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.

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.

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 X\subset\mathbb{R}^2 be a finite union of bounded polytopes and let T:X\to X be piecewise affine and eventually expanding. Then the Perron–Frobenius operator \mathcal{L} of T is quasicompact as an operator on the space of functions of bounded variation on \mathbb{R}^2 and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta function of T. In higher dimensions the result remains true under an additional generically satisfied transversality assumption.

Intuitively, a Liouvillian function on \mathbb{C} P(n) is one which is obtained from rational functions by a finite process of integrations, exponentiations and algebraic operations. This paper is devoted to the study of foliations determined by polynomial 1-forms which have a Liouvillian first integral. Our main result states that, under some mild restrictions on the singularities of the foliation, such a foliation must be either a linear foliation or an exponent two Bernoulli foliation after some rational pull-back. This proves that the highest level of transcendence for the ordinary differential equations which can be integrated by the use of elementary functions is reached at the Riccati equations.

We prove that the Jouanolou foliation of degree k ≤ 5 admits no non-trivial minimal sets. The proof uses reliable computations based on interval arithmetic.

We show that the strong stable foliation of the canonical extension of an Axiom A flow to an Abelian cover of a compact manifold is ergodic with respect to the product of a Gibbs measure on the manifold times the Haar measure on the fiber, as soon as it is transitive on the support of the measure. This generalizes previous results of M. Babillot, F. Ledrappier, M. Pollicott and V. Kaimanovich.

We study continuous cocycles defined on the set of planar tilings with values in discrete groups. Following Schmidt we show that for generalized domino tilings, L-tiles, and some systems of paths there exists a fundamental cocycle, i.e. we find a cocycle cf, so that all other continuous cocycles c are cohomologous to a homomorphic image of cf.

This paper gives a simple proof of the well-known fact that a codimension-one Anosov diffeomorphism of a closed manifold is topologically conjugate to a hyperbolic toral automorphism.

For a topologically mixing Anosov diffeomorphism on a compact manifold, the correlation function for two smooth functions is known to have exponential decay. As a generalization, higher correlation functions for several smooth functions are defined, and are shown to have exponential decay in the time variables. It is also proved that the higher derivatives of the pressure are equal to the summations of the higher correlations over the time variables.

The dynamics of a shift-like mapping f(x,y,z)=(y,z,yz+by+cz+dx+e) (d≠0) is studied, which is one of the normal forms of irregular quadratic polynomial automorphisms of [Copf ]3. It is shown that f has Green functions G±(x,y,z) associated to the dynamical degree of f. To obtain them, we consider a sort of filtration property.

Lorenz maps are dynamical systems on the interval related to the Lorenz flow. Monotone Lorenz families will be introduced and it will be shown that all possible topological dynamical behavior of Lorenz maps is realized within every monotone Lorenz family. Moreover, the structure of the parameter space for such families is described.

In the planar N-body problem, N point masses move in the plane under their mutual gravitational attraction. It is classical that the dynamics of this motion conserves the integrals of motion: center of mass, linear momentum, angular momentum c, and energy h. Further, the motion has a rotational symmetry. The dynamics thus takes place on a (4N-7)-dimensional open manifold, known as the reduced integral manifold [mfr ]R(M,ν). The topology of this manifold depends only on the masses M = (m1,...,mN) and the quantity ν = -hc2. In spite of the central importance of this manifold in a classical dynamical problem, very little is known about the topology of [mfr ]R(M,ν). In this note, we build on the topological analysis of Smale to describe the homology of [mfr ]R(M,ν). A variety of homological results are presented, including the computation of the homology groups for ν very large for all M; and for all ν for three masses, and for four equal masses.

We compute the homology of the integral manifolds of the restricted three-body problem—planar and spatial, unregularized and regularized. Holding the Jacobi constant fixed defines a three-dimensional algebraic set in the planar case and a five dimensional algebraic set in the spatial case (the integral manifolds). The singularities of the restricted problem due to collusions are removable, which defines the regularized problem.

There are five positive critical values of the Jacobi constant: one is due to a critical point at infinity, another is due to the Lagrangian critical points and three are due to the Eulerian critical points. The critical point at infinity occurs only in spatial problems. We compute the homology of the integral manifold for each regular value of the Jacobi constant. These computations show that at each critical value the integral manifolds undergo a bifurcation in their topology. The bifurcation due to a critical point at infinity shows that Birkhoff's conjecture is false even in the restricted problem.

Birkhoff also asked if the planar problem is the boundary of a cross section for the spatial problem. Our computations and homological criteria show that this can never happen in the restricted problem, but may be possible in the regularized problem for some values of the Jacobi constant. We also investigate the existence of global cross sections in each of the problems.

We study from a combinatorial point of view quadratic fibered polynomials. The main object of this paper is the construction of a combinatorial model of a subset of the parameter space. This set (the configuration space) is proved to be a Hausdorff compact and connected space. Then, we prove that an abstract configuration which is non-recurrent can be uniquely realized by a quadratic fibered polynomial.