We sharpen a result of Boyle, Marcus and Trow as follows. An aperiodic shift of finite type ΣA factors onto another ΣB with equal entropy by a 1-to-l almost everywhere right-closing map if and only if (1) the dimension group for ΣB is a quotient of that for ΣA; and (2) ΣA and ΣB satisfy the trivial periodic point condition for existence of a shift-commuting map from ΣA to ΣB.

For functions meromorphic in the plane, apart from an exceptional case, the Julia set J is the closure of the set of all preimages of poles. The repelling periodic cycles are dense in J. In contrast with the case of transcendental entire functions, J may be a subset of a straight line and general classes of functions for which this is the case can be determined. J may also lie on a quasicircle through infinity which is not a straight line.

The n-od is defined to be the set of all complex numbers z such that zn is a real number in the interval [0,1], i.e., a central point with n copies of the unit interval attached at their endpoints. Given a space X and a function f:X → X, Per (f) is defined to be the set {k: f has for a point of (least) period k, k a positive integer}. The main result of this paper is to give, for each n, a complete characterization of all possible sets Per (f), where f ranges over all continuous functions on the n-od.

Let X be a two-dimensional orientable connected manifold without boundary, H: T*X → ℝ a smooth hamiltonian function denned on the cotangent bundle. We will assume that H is of a ‘classical type’ that is convex and even on each fibre Tx*X. The goal of this paper is to describe the set Γ of all singular points of the projection Θ|L where ι: L → T*X is a smooth embedded 2-torus invariant under the hamiltonian flow h1, Θ: T*X → X is the canonical projection.

The introduction of results from harmonic analysis leads to new methods in the study of the ergodic properties of measures under the action of the direct sum of finite groups. We take the first steps in a systematic development of part of ergodic theory based on the formalism of the Riesz product construction.

We show the stability in the sense of α-congruence of dispersive (Sinai) planar billiards that are Bernoulli flows. The perturbations are either billiards on slightly altered tables or geodesic flows on nearby manifolds.

A simpler treatment is given of the theorems of Dye and Krieger concerning the classification of non-singular ergodic transformations up to orbit equivalence.

This paper is motivated by the connections between automorphisms of real suspension flows and ℝ2 suspension actions. Automorphisms which naturally lead to ℤ2-cocyles are examined from the viewpoint of covering theory in terms of an associated cylinder flow. A natural type of automorphisms (called simple) is analyzed via ergodic methods. It is shown that all automorphisms of suspensions built over minimal rotations on tori satisfy this condition. A more general approach using eigenfunctions extends this result to minimal affines, Furstenberg-type distal flows, certain nilmanifolds and a class of non-distal flows on the 2-torus.

In this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time-dependent potentials, namely, for systems of the form

where

is a superquadratic polynomial potential with periodic coefficients. As a limiting case, a proof of the stability of Ulam's problem of a particle bouncing between two periodicially moving walls is given.

We call nilmanifold every compact space X on which a connected locally compact nilpotent group acts transitively. We show that, if X is a nilmanifold and f is a continuous function on X, then, for all x in X and a in N, the sequence

converges. We give a process for the computation of the limit. A similar result for the continuous means is presented.

A slightly improved version of the (idealized) C1 closing lemma is proved. This turns out to generalize the C1 closing lemma from diffeomorphisms to nonsingular endomorphisms.