We show that the existence of a finitely summable unbounded Fredholm module (h, D) on a C* algebra A implies the existence of a trace state on A and that no such module exists on the C* algebra of a non amenable discrete group. Both for the needs of non commutative differential geometry and of analysis in infinite dimension we are led to the better notion of the θ-summable Fredholm module.

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.

The Ergodic Theory and Dynamical Systems journal pays tribute to V. A. Rokhlin, one of the founders of ergodic theory, a world-renowned topologist and geometer, and a man of tragic fate and exceptional courage. Rokhlin's mathematical heritage splits rather sharply into the ergodic theory – measure theory and topology – geometry parts. This fact has to do with a natural evolution of his interests but also with the keen sense of style in mathematics that Rokhlin possessed to an unusual degree. Naturally, we will concentrate on Rokhlin's contributions to ergodic theory and measure theory, his influence on other mathematicians working in those fields, and the development of some of his ideas. Fortunately, the topology part of Rokhlin's heritage has been superbly presented in Part I of the book ‘A la recherche de la topologie perdue’ published by Birkhaüser in Progress in Mathematics series (v. 62, 1986). The same cannot be said about the work on real algebraic geometry, Rokhlin's last big achievement. The impact of that work, carried out by his students, however, is very much felt now, and is very unlikely to be forgotten or neglected. On the other hand, his work in ergodic theory and measure theory, was restricted primarily to two relatively short time periods, 1947–1950 and 1959–1964, and for the most part was not followed by Rokhlin's immediate students. Hence, it runs a certain risk of being underestimated. We hope that the articles by A. Vershik; S. Yuzvinsky and B: Weiss, published in this issue, will put Rokhlin's work and his influence in ergodic theory into the right perspective.

We consider one-parameter families of torus diffeomorphisms that bifurcate from global hyperbolic maps (Anosov) to DA maps (derived from Anosov). For an open set of these families, we show that the Hausdorff dimension and limit capacity of the nonwandering set are not continuous across the bifurcation. We also study the behaviour of equilibrium measures near the bifurcation.

We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

Ergodic theory is one of several fields of mathematics where the name Vladimir Abramovich Rokhlin (spelled also ‘Rochlin’ and ‘Rohlin’) is well known to the specialists. That name is attached to some very often used theorems, but the goal of this paper is not just to remind the reader of these theorems. I put them in the context of the general development of ergodic theory during the thirty years 1940–70. Most of all, I want to emphasize that Rokhlin was not only a researcher producing powerful results but also a founder of schools at the two best Universities in the USSR. For at least 10 years (1958–68) these schools dominated ergodic theory. This paper is not biographical. Rokhlin's life certainly deserves a better biographer. However, I mention certain circumstances of a non-mathematical nature where it seems to be appropriate.

We study questions concerning the ergodic theory, von Neumann algebras, geometry, and topology of actions of lattices in Sp (1, n).

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.

The impact of V. A. Rokhlin's work in ergodic theory is discussed with particular emphasis on his famous lemma and its generalizations and his foundational work on measurable partitions.

We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a group Q which serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) of Q on it; in the case where Q is amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

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 study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to the K property of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

A new proof and a generalization of the Osserman-Sarnak estimate for the measure theoretic entropy of geodesic flows is presented.

The Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.

Vladimir Abramovich Rokhlin was born on 23 August 1919 in Baku (Azerbaijan). His parents, Abram Beniaminovich Rokhlin and Henrietta Emmanuilovna Levenson, came from Jewish families, who lived in the Ukraine and in Byelorussia and then moved to Baku. Rokhlin's mother was the sister of the well-known literary figure and children's writer Kornei Chukovsky; Rokhlin's grandmother, Klara Levenson, was one of the first women doctors in Russia. Rokhlin's mother graduated from a medical school in France and was a doctor in Baku. She died tragically in 1923. His father was a broadly educated man and took an active part in political activity before the revolution (he was a social democrat) and in the early years of the revolution. Later he was involved in administrative work in Baku, in the Ukraine, in Central Asia and in Moscow. Not surprisingly he did not escape the Stalinist repressions: in 1939 he was arrested and on 13.7.1941 was sentenced to be shot. In 1957 his relatives received a certificate of rehabilitation (‘the case is closed due to insufficient evidence’); it is clear from this certificate that it was still impossible to obtain reliable information about the last years of his life; in particular even the exact date of his death is not precisely established. Rokhlin's family was exiled to Siberia and remained there. Fortunately Rokhlin, who was at that time a student at the University of Moscow, escaped with comparatively minor unpleasantnesses, and was not expelled from the University.

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.

The Margulis measure for Anosov flows arises from a Hausdorff measure for a natural distance on unstable leaves. This generalizes work of Ursula Hamenstädt for the case of geodesic flows.

In this paper we will present a result which gives a sufficient condition for a vector field X on ℝ3 to be equivalent at a singularity to the first non-vanishing jet jkX(p) of X at p. This condition - which only depends on the homogeneous vector field defined by jkX(p) - is stated in terms of the blown-up vector field (which is defined on S2xℝ), and essentially means that there are no saddleconnections for |S2×{0}.

The key tool in the proof will be a result of local normal linearization along a codimension 1 submanifold M providing a C0 conjugacy having a normal derivative along M equal to 1.

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

Our purpose in this paper is to present a more or less complete solution to the problem of the smoothness of the conjugation of aperiodic diffeomorphisms of the circle. We show that the rotation number and the smoothness of the diffeomorphism guarantee a certain smoothness for the homeomorphism which conjugates it with a rigid rotation, and obtain the best smoothness that can be guaranteed.