A class of automorphisms of the unit square called generalized baker'stransformations (gbt) is defined in such a way that every stationary stochastic process may be represented as the movement of a simple partition of the square under a gbt. This extends the classical example of the representation of independent processes by the well-known baker's transformation.

Every ergodic, positive-entropy automorphism is measurably isomorphic to some gbt (again generalizing the classical result about Bernoulli shifts), and we show that a large class of gbt's satisfying certain continuity restrictions are actually measurably isomorphic to Bernoulli shifts.

It is shown that a C2 flow on a compact three-dimensional manifold that preserves a smooth measure and has a continuous family of cones satisfying a certain invariance property must be ergodic.

We obtain a family of metrics on the two-dimensional sphere whose geodesic flow is ergodic and Bernoulli. This family includes real analytic metrics.

We consider multi-applications Γ of the circle S1, the graphs of which are ‘degree (1, 1)’, continuous piece-wise monotonic curves of S1 × S1 In general Γp is not a connected curve but it is a union of a degree (1, 1) continuous curve Γp of S1 × S1 and of some other curves homotopic to a point. Using these Γp we are able to study dynamics of Γ. We focus on the case where Γ has no periodic points and we see, for instance, that all ‘regular’ orbits have, on S1 the same order as orbits of an irrational rotation. Using this we prove that such F without ‘cycles’ are obtained from a Denjoy's counter-example, perturbing it in the holes of the invariant set. Finally we generalize the classical result of Block and Franke showing that if Γ is a C2 curve with no degenerate critical points, or if Γ is a C∞ curve with no ‘flat’ points, there are always ‘cycles’, unless Γ is an homeomorphism.

This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.

We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).

So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?

Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

Let T be a translation defined on a nil-manifold N/Γ, where N is a nilpotent group of order two.

Suppose we are given an action α: G → Aut (M) of a group G on a factor M; α possible way to analyse α may be to look at the invariant components where the action becomes more tractable. This point of view naturally leads to the study of the injective invariant subalgebras (recall for instance the good properties shared by amenable discrete or compact actions in the hyperfinite case [14]).

We consider inner functions on the unit disk which have a finite number of singularities on the unit circle. The restriction of such functions to the circle are maps onto the circle. We give sufficient conditions that these restrictions are exact endomorphisms whose natural extensions are Bernoulli and that the entropy is given by Rohlin's formula, We also give the entropy in closed form if ƒ' is in the Nevalinna class N. An example is considered. In the last section we show that if two restrictions are metrically isomorphic, they are diffeomorphic.

Letf: M → M be a C1 map on a compact manifold. We give a topological condition under which f has an even number of periodic points with a given period. We also obtain a sufficient condition, in terms of homology, for ƒ to have infinitely many periodic points.

We describe ergodic properties of the system of two hard discs with arbitrary masses moving on the two dimensional torus.

For certain C2 one-parameter families of endomorphisms of the circle both endpoints of rotation intervals are rational except for a set of parameter values of zero Lebesgue measure.

M. Ratner's theorem on the rigidity of horocycle flows is extended to the rigidity of horospherical foliations on bundles over finite-volume locally-symmetric spaces of non-positive sectional curvature, and to other foliations of the same algebraic form.