Let M be a two-dimensional, compact manifold and g:Μ→ΜM be a diffeomorphism with a hyperbolic chain recurrent set. We find restrictions on the reduced zeta function p(t) of anyzero-dimensional basic set of g. If deg (p(t)) is odd, then p(1) = 0 (in ). Since there are infinitely many subshifts of finite type whose reduced zeta functions do not satisfy these restrictions, there are infinitely many subshifts which cannot be basic sets for any diffeomorphism of any surface.

In this paper we construct an example of a homeomorphism of the circle onto itself which is C∞, has no periodic points and no dense orbits. Moreover, the homeomorphism will have no more than two points of zero derivative. We alter this example to form a C∞ map of an interval to itself which has homtervals.

There exists a Bernoulli shift with non-identical factor measures for which no invariant σ-finite equivalent measure exists.

If p and q are probability vectors of equal entropy each having at least three non-zero components then there exists a finitary homomorphism between the corresponding one-sided Bernoulli shifts.

We consider a certain analytic function β (t) which is an invariant of finite equivalence between two finite state Markov chains. If two such chains P, Q have the same β-function we wish to prove that they are finitely equivalent. To this end we show that U(t)Pt = QtU (t) has a nontrivial matrix solution U over the ring (exp) of integral combinations of exponential functions. In fact we can force U(t) to be strictly positiveat any specified t0. If U(t) has entries from (exp), the sub-semi-ring of positive integral combinations of exponential functions, then P, Q are finitely equivalent. Many examples reinforce the conjecture that U(t) may always be chosen over (exp) when P, Q have the same β-function. We relate the β-function to topological entropy, measure entropy and information variance.

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.

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 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.