In this article we consider conjugacies between differentiable expanding maps. We give a necessary and sufficient condition for a conjugating map to be absolutely continuous, in terms of the Jacobian around periodic orbits.

Let N be the class of meromorphic functions f with the following properties: f has finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points of f are zeros of f′ and vice versa, with finitely many exceptions; f has finite order. It is proved that if f ∈ N, then f does not have wandering domains. Moreover, if f ∈ N and if ∞ is among the limit functions of fn in a cycle of periodic domains, then this cycle contains a singularity of f−1. (Here fn denotes the nth iterate of f) These results are applied to study Newton's method for entire functions g of the form where p and q are polynomials and where c is a constant. In this case, the Newton iteration function f(z) = z − g(z)/g′(z) is in N. It follows that fn(z) converges to zeros of g for all z in the Fatou set of f, if this is the case for all zeros z of g″. Some of the results can be extended to the relaxed Newton method.

We study the problem of lifting ergodicity to certain skew-product extensions over shifts of finite type. This can be done by computing the asymptotic range of the defining cocycle. For a class of ergodic shift-invariant measures and a class of functions (defining conservative extensions) we give a characterization of ergodicity in terms of weights of periodic orbits in the shift space.

We give necessary conditions for an endomorphism of the interval not to be C2+α-approximated by endomorphisms having a link between critical points. We apply these techniques to show that on a C2+α-generic set of unimodal maps the endomorphisms are either Axiom A or there exists an ergodic measure with zero Lyapunov exponent. Applying this result we prove the formula conjectured by Eckmann and Ruelle for the rate of escape of the complement of the basin of the sinks. We add some remarks on the C2-stability conjecture.

A structure theorem is established for amenable actions of a countable discrete group.

We prove that there exists a constant δ > 0 such that for almost every pair of numbers α and β there exists n0= n0(α,β) such that for any n ≥ n0

where the integers pn,qn, rn are provided by the modified Jacobi-Perron algorithm.

Fried has related closed orbits of the geodesic flow of a surface S of constant negative curvature to the R-torsion for a unitary representation of the fundamental group of the unit tangent bundle T1S. In this paper we extend those results to transitive Anosov flows and 2-dimensional attractors on 3-manifolds.

In this note, we show the connectedness of the tricorn, the connectedness locus for the family of antiquadratic maps: fc(z) = + c, c ∈ C.

Recently H. Hofer defined a new symplectic invariant which has a beautiful dynamical meaning. In the present paper we study this invariant for Lagrangian submanifolds of symplectic manifolds. Our approach is based on Gromov's theory of pseudo-holomorphic curves.

We consider two-dimensional diffeomorphisms φ: M → M. For a fixed point p, i.e., p ∈ M and φ(p) = p, we say that q is homoclinic to p if p ≠ q and if both limi→+∞φi(q) and limi→−∞φi(q) are equal to p.

We consider transitive Anosov flows φ: M → M and give necessary and sufficient conditions for every homology class in H1(M,ℤ) to contain a closed φ-orbit. Under these conditions, we derive an asymptotic formula for the number of closed φ-orbits in a fixed homology class, generalizing a result of Katsuda and Sunada.

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.