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.

We are interested in ergodic properties of absolutely continuous invariant measures of positive entropy for a map of an interval. We prove a Bernoulli property and a characterization by some variational principle.

We construct C1 symmetric maps of [−1, 1] to itself, satisfying a condition similar to Feigenbaum's: -φ(x) = φ(-x), φ2(bx) = bφ(x), 0<b <1. Under certain conditions, the non-wandering set consists of the one-sided Morse minimal set together with 2n points of period 2n for each n = 1, 2, … The main significance of the construction is its simplicity. Given a certain piece of the map φ, the rest is generated by the required equation.

Let Г be a finitely generated non-elementary Fuchsian group acting in the disk. With the exception of a small number of co-compact Г, we give a representation of g ∈ Г as a product of a fixed set of generators Гo in a unique shortest ‘admissible form’. Words in this form satisfy rules which after a suitable coding are of finite type. The space of infinite sequences Σ of generators satisfying the same rules is identified in a natural way with the limit set Λ of Г by a map which is bijective except at a countable number of points where it is two to one. We use the theory of Gibbs measures onΣ to construct the so-called Patterson measure on Λ [8], [9]. This measure is, in fact, Hausdorff 5-dimensional measure on Λ, where S is the exponent of convergence of Г.

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.

We associate to each complex simple Lie algebra g a hierarchy of evolution equations; in the simplest case g = sl(2) they are the modified KdV equations. These new equations are related to the two-dimensional Toda lattice equations associated with g in the same way that the modified KdV equations are related to the sinh-Gordon equation.

Let f: M → M be a diffeomorphism of a compact manifold M and let χ:M → R be defined by putting χ(x) equal to the sum of the non-negative characteristic exponents of f at x, each being counted with its multiplicity. If μ is an f-invariant probability of M which is absolutely continuous relative to Lebesgue measure, then Pesin has proved the entropy, hμ(f), is given by We prove this formula without using the theory of stable manifolds.

Let Г be a finitely generated discrete subgroup of the isometries of the hyperbolic plane H2 with at least one parabolic element. We prove that, if Г1 is a subgroup of Г with Г/Г1 abelian, the ‘critical exponent’ of Г1 is the same as that of Г. We give necessary and sufficient conditions-in terms of the rank of Г/Г1, the critical exponent of Г, and the image of parabolic elements of Г in Г/Г1 - for the Poincaré series of Г1 to diverge at the critical exponent.

It is proved that for the map f: ℂ → ℂ given by f(z) = ez, the family is not normal at any point. In particular, f is topologically transitive.

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.

Let f be a diffeomorphism of a manifold and Λ be an f-invariant set supporting an ergodic Borel probability measure μ with certain properties. A lower bound on the capacity of Λ is given in terms of the μ-Lyapunov exponents. This applies in particular to Axiom A attractors and their Bowen-Ruelle measure.

This paper discusses the relations between the following properties o finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definations these notions lead to some interesting examples.

This paper concerns a problem which arose from a paper of Sullivan. Let Γ be a discrete group of isometries of hyperbolic space Hd+1. We study the question of when the geodesic flow on the unit tangent bundle UT (Hd+1/Γ) of Hd+1/Γ is ergodic with respect to certain natural measures. As a consequence, we study the question of when Γ is of divergence type. Ergodicity when the non-wandering set of UT (Hd+1/Γ) is compact is already known from the theory of symbolic dynamics, due to Bowen, or from Sullivan's work. For such a Γ, we consider a subgroup Γ1 of Γ with Γ/Γ1 ≅ℤυ and prove the geodesic flow on UT (Hd+1/Γ1) is ergodic (with respect to one of these natural measures) if and only if υ ≤ 2.

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.

The rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.