We prove the rational egodicity of geodesic flows on divergence type surfaces of constant negative curvature, and identify their asymptotic types.

Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This paper studies the structure of simple permutations which have order a power of two, where simple permutations are permutations corresponding to the simple orbits of Block.

The property of a one-parameter C*- (or W*-) dynamical system that the spectral subspaces corresponding to the three subsets (−∞, 0), {0}, and (0, +∞) add up to the whole algebra is reformulated. If the C*-algebra is prime (or the W*-algebra is a factor), an equivalent property is that the spectrum is finite.

We prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.

Poincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

Given a topological process (X, µ, T) where T is a homeomorphism of the compact metric space X which preserves the probability measure µ and is ergodic, we show that there exists an uncountable family {(Xi, µi, Ti)}i∈I of topological processes such that for every i, (Xi, µi, Ti) is measure-theoretically isomorphic to (X, µ, T) but for every i ≠ j, (Xi, µi, Ti) and (Xj, µj, Tj) are not almost topologically conjugate.

This paper addresses the following long-standing open question: If a stationary transformation on a probability space obeys the property

for all measurable sets A1, A2, does it follow that

for all measurable sets A1, A2, A3? Here we answer the question affirmatively for a certain class of transformations.

The fractal dimension of an attracting torus Tk in × Tk is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1

We give an algebraic characterization of the class of spectral radii of aperiodic non-negative integral matrices, and describe a method of constructing all such matrices with given spectral radius. The logarithms of the numbers in are the entropies of mixing topological Markov shifts. There is an arithmetic structure to , including factorization into irreducibles in only finitely many ways. This arithmetic structure has dynamical consequences, such as the impossibility of factoring the p-shift into a direct product of nontrivial homeomorphisms for prime p.

The dynamical system associated to the difference equation

has been studied numerically by several authors. On the basis of numerical evidence, they conclude that there exists a number k0 ≈ 0.97 such that there are homotopically non-trivial invariant circles for |k|≤k0 and there are none for |k|>k0. In this note, we give a simple rigorous proof that there are none for |k|>.

Examples are constructed of ergodic rational maps with dense critical orbit.