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.

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.

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 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.

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.

This paper discusses a division of one-dimensional cellular automata into classes which seem to correspond to different types of behaviour.

Ruelle and Capocaccia gave a new definition of Gibbs states on Smale spaces. Equilibrium states of suitable functions thereon are known to be Gibbs states. The converse is discussed in this paper, where the problem is reduced to shift spaces and solved there by constructing suitable conjugating homeomorphisms in order to verify the conditions for Gibbs states which Bowen gave for shift spaces, where the equivalence to equilibrium states is known.

According to M. Gromov, any sequence of Riemann manifolds with uniformly bounded geometry has a subsequence that converges to a limit. It is shown here that this limit Riemann structure is Lipschitz, generates a Lipschitz geodesic flow, and consequently, as Gromov asserted, the limit distance function is of class C1,1. Sharpness of the results is discussed. A simple, extrinsic proof of Gromov's Theorem is included.

Confirming a conjecture by Boshernitzan, it is proved that if T is a minimal non-uniquely ergodic interval exchange, the minimum spacing of the partition determined by Tn is O(1/n).