We prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

We consider some very simple examples of SL(2, ℝ)-cocycles and prove that they have positive Lyapunov exponents. These cocycles form an open set in the C1 topology.

Let (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfies

where ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

We define a topology, denoted by Cω, on the space of real analytic diffeomorphisms and show that the elements of a residual (Baire second category) subset of diffeomorphisms that satisfy Axiom A and the Transversality Condition have a trivial centralizer; they only commute with their integer powers.

Some conditions are given under which the Hausdorff dimension of an attractor of an endomorphism of a surface can be determined using the pressure formula.

We give constraints on the existence of factor maps between sofic shifts. These constraints yield examples of sofic shifts of entropy log n which do not factor onto the full n-shift. We also show that any prime which divides the degree of an endomorphism of a sofic shift must divide the non-leading coefficients of the characteristic polynomial of the core matrix of the shift.