The notion of an iterated extension of a flow is introduced and studied. In particular it is shown how eigenfunctions occur in a natural way. This is then exploited to produce an example of a weakly mixing minimal set with a non-weakly mixing quasi-factor.

We determine when a polynomial is the reduced zeta function of a basic set of a Smale diffeomorphism of a compact surface.

We show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:

(i) sup φ — inf φ <htop(T) and φ is of bounded variation.

(ii) φ satisfies a variation condition and T has a local specification property.

(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

For an ergodic transformation T, necessary and sufficient conditions are given for a set A and an integer n so that the existence of a set B with pairwise disjoint and is ensured.

We study the dynamical properties of ergodic toral autmorphisms that have some eigenvalues of modulus one. For such automorphisms, all sufficiently fine smooth partitions generate measurably, but never topologically, and are never weak Bernoulli. The points of period k become uniformly distributed exponentially fast, and Lipschitz functions mix exponentially fast. Every reasonably smooth compact null set has the property that there is a dense set of periodic points whose entire orbit misses the set, but this is false for general compact null sets. Katznelson's property of almost weak Bernoulli can be strengthened to a certain exponential rate of independence, but breaks down at a critical number. Finally, open sets have return times that decay exponentially fast.

A connection between the symbolic description of the geodesic flows on certain modular surfaces and the theory of continued fractions is developed. The well-known properties of these dynamical systems then lead to some new results about continued fractions.

We here show that the necessary condition of exponentially decaying first return times to a cylinder set is also sufficient for a countable state Markov chain to be almost continuously isomorphic to a Bernoulli shift.

The purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see corollary 5).

We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.