The totally minimal flow (X, T) is said to have finite almost periodic rank if there is a positive integer n such that whenever (x1, x2,…, xn+1) is an almost periodic point of the product flow (Xn+1, T×…×T) then, for some i≠j, xi, and xj are in the same orbit. The rank of (X, T) is the smallest such integer. If (Y, S) is a graphic flow, (Y, Sn) has rank |n| and it is shown that every finite rank flow has, modulo a proximal extension, a graphic power factor. Various classes of finite rank flows are defined, and characterized in terms of their Ellis groups. There are four disjoint types which have basic structural differences.

When considering an action α of a compact group G on a C*-algebra A, the notion of an α-invariant Hilbert space in A has proved extremely useful [1, 4, 8, 14, 17, 18]. Following Roberts [13] a Hilbert space in (a unital algebra) A is a closed subspace H of A such that x*y is a scalar for all x, y in H. For example if G is abelian, and α is ergodic in the sense that the fixed point algebra Aα is trivial, then A is generated as a Banach space by a unitary in each of the spectral subspaces

which are then invariant one dimensional Hilbert spaces. If G is not abelian, then Hilbert spaces (which are always assumed to be invariant) do not necessarily exist, even for ergodic actions. For non-ergodic actions, it is also desirable to relax the requirement to x*y being a constant multiple of some positive element of Aα. More generally, if γ is a finite dimensional matrix representation of G and n is a positive integer, we define to be the subspace

where d is the dimension d(γ) of γ, and Mnd denotes n×d complex matrices. (Usually we will denote the extended action of αg to αg ⊗ 1 on A⊗Mnd again by αg.) Let .

If the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

Let ƒ denote a continuous map of a compact interval I to itself. A point x ∈ I is called a γ-limit point of ƒ if it is both an ω-limit point and an α-limit point of some point y ∈ I. Let Γ denote the set of γ-limit points. In the present paper, we show that (1) −Γ is either empty or countably infinite, where denotes the closure of the set P of periodic points, (2) x ∈ I is a γ-limit point if and only if there exist y1 and y2 in I such that x is an ω-limit point of y1, and y1 is an ω-limit point of y2, and if and only if there exists a sequence y1, y2,…of points in I such that x is an ω-limit point of y1, and yi is an ω-limit point of yi+1 for every i ≥ 1, and (3) the period of each periodic point of ƒ is a power of 2 if and only if every γ-limit point is recurrent.

We are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

We study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.

In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.

In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

In this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.