Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied.

Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.

We characterize the algebraic-geometric potentials for the Schrödinger and AKNS operators using the Weyl m-functions and the Floquet exponent for these operators. The characterization is this: among random ergodic Schrödinger operators, the alebraic-geometric potentials are those for which (i) the spectrum is a union of finitely many intervals (or bands); (ii) the Lyapounov exponent vanishes on the spectrum.

We present a dynamical system in ℝ2 with a global point attractor but so that two such systems, when coupled by linear diffusion, produce a system in ℝ4 with no point attractors and yet with all solutions bounded in the positive time direction.

Consider a Markov process on a locally compact metric space arising from iteratively applying maps chosen randomly from a finite set of Lipschitz maps which, on the average, contract between any two points (no map need be a global contraction). The distribution of the maps is allowed to depend on current position, with mild restrictions. Such processes have unique stationary initial distribution [BE], [BDEG].

We show that, starting at any point, time averages along trajectories of the process converge almost surely to a constant independent of the starting point. This has applications to computer graphics.

We extend results of Bowen and Manning on systems with good symbolic dynamics. In particular we identify the class of dynamical systems that admit Markov partitions. For these systems the Manning-Bowen method of counting periodic points is explained in terms of topological coincidence numbers. We show, in particular, that an expansive system with a finite cover by rectangles has a rational zeta function.

Relationships between the permutations associated to periodic orbits of maps of the interval and the topology of the space of these maps are given. These extend the author's previous work on periodic orbits to orbit types.

It is shown that when viewed properly some concepts in topological dynamics and ergodic theory are not merely analogous but equivalent. Also the Mackey-Halmos-von Neumann theorem on ergodic processes with discrete spectrum is generalized and an account of the Mackey-Zimmer theory of minimal cocycles is given in a more general setting.

Suppose that (X, ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of (X, ℬ, μ). T is called weakly mixing if for any two sets A1A2 ∈ ℬ one has:

Let M be a compact manifold that contains a two-dimensional punctured torus. Given p ∈ M and an integer r ≥ 2, there exists X ∈ ∞(M) having non-trivial recurrent trajectories and such that, for some neighbourhood of X|(M−{p}) in r(M−{p}), no Y ∈ has closed orbits.

We consider the Markov chain on a compact manifold M generated by a sequence of random diffeomorphisms, i.e. a sequence of independent Diff2(M)-valued random variables with common distribution. Random diffeomorphisms appear for instance when diffusion processes are considered as solutions of stochastic differential equations. We discuss the global dynamics of Markov chains with continuous transition densities and construct non-random stable foliations for random diffeomorphisms.

We develop simple topological criteria for the existence of periodic orbits in maps of the annulus. These are applied to one-parameter families of dissipative twist maps of the annulus and their attractors. It follows that many of the motions found by variational methods in area preserving twist maps also occur in the dissipative case.

Let g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: g → g′. Furthermore, any homeomorphic conjugacy g → g′; must be a C1 diffeomorphism.

We call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings. This generalizes Rudolph's notion of minimal self-joinings and Veech's property S.

Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed. A weakly mixing group extension of an action with minimal self-joinings is simple. The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple. Some corollaries. Two simple actions with no common factors are disjoint. The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a -action with minimal self-joinings are disjoint.

An extension from Riemann to Lorentz manifolds is proved for Anosov's theorem on the hyperbolicity (or exponential sensitivity to initial conditions) of the geodesic flow.

A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate to

We define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent to

Our main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to

We prove that the horospherical foliations of two compact manifolds of constant negative curvature are measurably isomorphic if and only if the two manifolds are isometric.

We construct a minimal homeomorphism of a compact space such that a sequence of its iterates converges pointwise to the identity but no sequence of its iterates converges uniformly.

A compact and isolated invariant set of a continuous flow possesses a so called Conley index, which is the homotopy type of a pointed compact space. For this index a well known continuation property holds true. Our aim is to prove in this context a continuation theorem for the invariant set itself, using an additional structure. This refinement of Conley's index theory will then be used to prove a global and topological continuation-theorem for normally hyperbolic invariant sets.

Let A be an n × n real matrix; sufficient conditions were previously worked out, assuming non-commensurability of eigenvectors, for A to be SL(n, )-conjugate to a matrix all sufficiently large powers of which have strictly positive entries. We show that when the ‘large’ eigenvectors are commensurable and satisfy the obvious necessary conditions, then A is also going to be so conjugate. In particular, we deduce, if A is a rational matrix with large eigenvalue exceeding 1 and of multiplicity one, then A is algebraically shift equivalent to an eventually positive matrix, using only integer rectangular matrices.

This paper generalises the well known structural stability theorem which says that a diffeomorphism is structurally stable if it is Axiom A and if all invariant manifolds are transversal to each other.

If these transversality conditions are not satisfied then the diffeomorphism not only fails to be stable, but also this gives rise to the appearance of moduli. That is, one needs several real parameters to parameterise all conjugacy classes of nearby diffeomorphisms. (The minimum number of parameters needed is called the number of moduli).

Here we deal with diffeomorphisms on two dimensional manifolds, whose asymptotic dynamics are well understood (the class of Axiom A diffeomorphisms). The main result characterises those Axiom A diffeomorphisms which have a finite number of moduli. This result can be regarded as a generalisation of the structural stability theorem. From the proofs it follows that the dynamics of these diffeomorphisms can also be well understood.

In the proof of our main theorem we need certain invariant foliations to be quite smooth. In an appendix we prove a differentiable version of the Lambda Lemma.