We classify finite-to-one factor maps between shifts of finite type up to almost topological conjugacy.

Let f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

This paper concerns Smale diffeomorphisms of compact oriented surfaces. Relationships are found between the isotopy class of the map and the dynamics of its basic sets. The form of the dynamical properties involves restrictions on periods and reduced zeta functions.

In this paper we show that Birkhoff periodic orbits actually exist for arbitrary monotone twist maps satisfying the graph intersection property.

We study which algebraic integers λ ≥ l arise as the growth rate of a mapping class of a surface and give conditions that are necessary and perhaps sufficient. Flow equivalence and twisted Lefschetz zeta functions are used to generate families of λ's. Examples and open problems are included

It is proved, using Krieger's theorem, that ITPFI's of bounded type are ITPFI2. This answers a question asked by E. J. Woods.

We introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flow.

We give examples of maps of the interval with zero entropy for which the continuous invariant measure has no dimension, and we prove a dimension property for maps lying in the stable manifold of Feigenbaum's fixed points.

Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.

A mixing transformation is constructed which has non-simple spectrum of finite multiplicity. This example is based on a rank 1 mixing transformation and is constructed by cutting and stacking. It can be made to be mixing of all orders.

We prove an almost sure invariance principle and a central limit theorem for the process , where f is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation (p ≥ 1).

The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process .