We study minimal flows and their extensions by means of the associated maximally highly proximal flows. These, in turn, can be represented by highly proximal generators, which are certain subsets of the universal minimal flow. From this point of view we obtain information on relative disjointness, coalescence, the Bronstein property, and RIC extensions.

Let f be a continuous map of the circle to itself. Let P(f) denote the set of periods of the periodic points. In this paper the set P(f) is studied for functions without fixed points, so 1∉P(f). In particular, it is shown that if s, t are the two smallest integers in P(f) and s and t are relatively prime then αs+βt∈P(f) for any positive integers α and β.

Let α be an automorphism of a finite von Neumann algebra and let H(α) be its Connes-Størmer's entropy. We show that H(α) = 0 if α is the induced automorphism on the crossed product of a Lebesgue space by a pure point spectrum transformation. We also show that H is not continuous in α and we compute H(α) for some α.

We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set:

It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.

For an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

We consider measured foliations on surfaces, and interval exchanges. We give alternative proofs of the following theorems first proved by Masur and (independently) Veech. The action of the diffeomorphism group of the surface on the projective space of measured foliations (with respect to a natural ‘Lebesgue’ measure) is ergodic. Almost all measured foliations are uniquely ergodic. Almost all interval exchanges (again, with respect to a natural ‘Lebesgue’ measure) are uniquely ergodic.

It is shown that for a lattice Γ in a semi-simple group of real rank 1 the action on the boundary always admits an equivariant topological Γ-factor. We also show that there are no such factors for SL (n, ℤ) acting on ℙn−1, n≥3.

A method is exposed which allows the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G. This is applied to invariant Hamiltonian systems on the tangent bundles of certain homogeneous spaces M = G/K. It is proved, for example, that every such invariant Hamiltonian system is completely integrable if M is a real or complex Grassmannian manifold or SU(n + 1)/SO(n + 1) or a distance sphere in ℂPn+1. In particular, the geodesic flows of these homogeneous spaces are integrable.

The point of this note is to answer in the affirmative a question of G. A. Margulis. In the course of his proof of the finiteness of either the cardinality or the index of a normal subgroup of an irreducible lattice in a higher rank semi-simple Lie group [3], [4], Margulis proves that if Γ = SL (n, ℤ), n≥3, (X, μ) is a measurable Γ-space, μ quasi-invariant, and φ: ℙn−1→X is a measure class preserving Γ-map, then either φ is a measure space isomorphism or μ is supported on a point. Margulis then asks whether the topological analogue of this result is true. This is answered in the following.