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.

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

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.

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

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.

By analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X, T) according to the various ways a sequence Tni can tend to the identity transformation. The main results obtained are:

(i) On a rigid flow there exists a T-invariant, symmetric, closed relation Ñ such that (X, T) is uniformly rigid iff Ñ = Δ, the diagonal relation.

(ii) For syndetically distal (hence distal) flows rigidity is equivalent to uniform rigidity.

(iii) We construct a family of rigid flows which includes Körner's example, in which Ñ exhibits various kinds of behaviour, e.g. Ñ need not be an equivalence relation.

(iv) The structure of flows in the above mentioned family is investigated. It is shown that these flows are almost automorphic.

It is a general problem to find appropriate sets of moduli for families of functions that generate dynamical systems. In this paper we solve this problem for a specific family of holomorphic self-maps of ℂ* defined by

The main theorem states that any function topologically conjugate to a member of ℱ is holomorphically conjugate to some member of the family. It follows that the coefficients of the polynomials P(z) and Q(z) are a suitable set of moduli for the families of dynamical systems generated by these functions.

The moduli spaces of functions in ℱ are easy to study computationally and have been studied by many authors. (See references in the text.)

Given a manifold equipped with a free, properly discontinuous, cocompact ℤ-action, and a flow on which is ℤ-equivariant, we study the qualitative dynamics of . Under certain hypotheses on , we show that the chain recurrent set of has a decomposition which is the analogue, in the category of ℤ-equivariant flows, of Smale's spectral decomposition for recurrent sets of Axiom A flows.

It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.

In this paper we study some conjugacy invariants (moduli) for discrete two dimensional dynamical systems, with a homoclinic tangency. We show that the modulus obtained by Palis in the heteroclinic case also turns up in the case considered here. We also present two new conjugacy invariants.