Let M be a rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C1 curve in the unit tangent sphere at a point p ∈ M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension 1.

A class of locally isometric, but not necessarily invertible mappings of an interval is considered. We show that under some conditions the study of the dynamical properties of these mappings can be reduced to interval exchange transformations. On the other hand, there are examples of mappings in this class with ergodic invariant measures supported by Cantor sets. The so-called μβ -sets studied by Y. Katznelson appear naturally in such examples.

A degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

A self-contained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ‘strong diophantine property’ hypothesis used previously.

It is shown that for certain classes of S-unimodal maps with aperiodic kneading sequences, the topological conjugacies are also quasisymmetric. This includes some infinitely renormalizable polynomials of unbounded type.

If (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

Almost-sure convergence of (l/k) log Wk(x, y) to entropy for weak Bernoulli processes is proved, where Wk (x, y) is the waiting time until an initial segment of length k of a sample path x is seen in an independently chosen sample path y. Analogous almost-sure results are obtained in the approximate match case for very weak Bernoulli processes. The weak Bernoulli proof uses recent results obtained by the authors about the estimation of joint distributions, while the very weak Bernoulli result utilizes a new characterization of such processes in terms of a blowing-up property.

It is shown that for a class of simple AF-algebras generated by a sequence of self-adjoint unitaries (sn) the entropy of the shift α(sn) = sn+1 with respect to the unique invariant trace υ satisfies hϕøϕ (α⊗α) = log2, while hφ(α) = 0.

We classify the C1+α structures on embedded trees. This extends the results of Sullivan on embeddings of the binary tree to trees with arbitrary topology and to embeddings without bounded geometry and with contact points. We used these results in an earlier paper to describe the moduli spaces of smooth conjugacy classes of expanding maps and Markov maps on train tracks. In later papers we will use those results to do the same for pseudo-Anosov diffeomorphisms of surfaces. These results are also used in the classification of renormalisation limits of C1+α diffeomorphisms of the circle.

We prove that the restriction of the graph of a subfactor, ΓN,M, to an infinite subset of vertices with finite boundary has the same norm as ΓN,W. In particular, if N φ M is extremal with [M : N] > 4 and ΓN,M has an A∞, tail then ΓN, M = A∞.