Let p = {p1,  , pn} and q = {qi,  , qm} be finite probability vectors, each having at least three non-zero components, such that . Let C = {1,  , n}, D = {1,  , m} and let (Cℤ, pℤ, σ) and (Dℤ, qℤ, σ) be the corresponding Bernoulli shifts. Then there exists an isomorphism φ between these shifts such that for a.a. x ∈ Cℤ σ(x)(0) is determined by finitely many of the future co-ordinates x(0), x(1),   and for a. a.y ∈ Dℤφ−1(y)(0) is determined by finitely many of the co-ordinates   y(−1), y(0), y(1),  .

If (Ω, μ, T) is a measure preserving system and if ƒ ∈ L1(μ) then, for almost all ω, for every real polynomial P, the sequence

To prove this result we use the concept of disjointness of dynamical systems and the fact that disjointness is preserved by isometric extensions.

Let S be a subgroup of a topological group T, and suppose that S acts on a space X. One can form a T-transformation group (X ×sT, T) called the suspension of the S-transformation group (X, S). In this paper we study the relationship between the dynamical properties of (X, S) and those of its suspension when S is syndetic in T. The main tool used in this study is a notion of the group of a minimal flow (X, T) which is sensitive to the topology on the group T. We are able, using this group and the enveloping semigroup to obtain results on which T-transformation groups can be realized as suspensions of S-transformation groups, and give conditions under which the suspension of an equicontinuous S-flow is an equicontinuous T-flow.

We study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given.

Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:

(1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit.

(2) There is a constant λT such that for Lebesgue-a.e. x.

(3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy.

(4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.

We prove that the lowest upper bound for the number of limit cycles of small nonconservative polynomial deformations of degree n of the Hamiltonian vector field is n − 1.

A consequence is that the lowest upper bound for the number of limit cycles of generic n-parameter deformations of cusps is n − 1.

We consider families of differentiable diffeomorphisms with hyperbolic points, close to the identity, which tend to it when the parameter goes to zero.

We study the asymptotic behaviour of the invariant manifolds. Then we consider the case when there are homo-heteroclinic points and we find that the maximum separation between the invariant manifolds is of the order of some power of the parameter which is related to the degree of differentiability.

Finally the analogous case for flows is considered.

Let denote the class of transformations with the weak Pinsker property, and let []m denote the class of transformations m-equivalent to some member of , where m is an entropy-preserving size. We show that if T is a factor of an element of []m, then T ∈ []m, and if T is an m-limit of elements of []m, then T ∈ []m.

Andrés del Junco has proposed a definition of topological minimal self-joinings intended to parallel Dan Rudolph's measure-theoretic concept. By means of a rank-two ‘cutting and stacking’, this article constructs the first example of a system (a subshift) satisfying his proposed definition of 2-fold topological minimal self-joinings.

The second part of the article shows that 2-fold topological minimal self-joinings does not imply 3-fold and that no map has 4-fold topological minimal self-joinings. This latter result follows from a generalization of a theorem of Schwartzman.

The purpose of this research is to describe all smooth vector fields on the torus T2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T2 which have non-trivial recurrence and also sinks.

We prove the meromorphicity of the zeta function on shifts of finite type for Hölder continuous functions assuming that the essential spectrum of the associated Ruelle operator is contained in the open unit disc. This result allows to extend the region of meromorphicity of the zeta function for Axiom A flows by a strip whose width is determined by the contraction rate of the flow.

We give a classification theorem for cellular automata, showing that either there is a minimal quasi-attractor whose basin has full measure, or else no chain component has a basin with positive measure.

Let (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

Let C denote the space of continuous functions mapping [0,1] into itself and endowed with the sup metric. It has been shown that C2 = {f ∘ f: ∈ C} is an analytic but non-Borel subset of C. This implies that there is no simple geometric characterization for a function being a square. In this paper we consider the problem of characterizing those functions which can be approximated by squares. In the first section we prove that any continuous function mapping a closed proper subset of [0,1 ] into [0,1 ] can be extended to a square. In particular this shows that C2 is Lp dense in C. On the other hand, C2 does not contain a ball when C is endowed with the sup metric. In the second section we prove that no strictly decreasing function can be uniformly approximated by squares, although the distance between the class of strictly decreasing functions and C2 is zero. In the last section we investigate the function f(x) = 1 − x and show that for every g ∈ C and that ¼ cannot be improved.

We define open index pairs of an isolated invariant set, prove their existence and compute the fixed point index of an isolating neighbourhood in terms of the Lefschetz number of a certain map associated with the open index pair. We use this to establish rationality of zeta functions and Lefschetz zeta functions.

For ergodic group extensions of transformations with discrete spectra it is proved that each invariant sub-σ-algebra is determined by a compact subgroup in the centralizer of a natural factor.

Suppose q is a holomorphic quadratic differential on a compact Riemann surface of genus g ≥ 2. Then q defines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at most T is at most quadratic in T. An application is given to billiards.

It is shown that the system of the form x + V′ (x) = p (t) with periodic V and p and with (p) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V ∈ C(5) and p ∈ L1; furthermore, for any solution x(t) there exists a velocity bound c for all time: |x(t)| < c for all t ∈ R. For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.

The Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

We give a classification of all the C*-algebras of Morse-Smale flows on closed two-manifolds, and determine the relation between the invariants of dynamical systems and the topological invariants of the C*-algebras.