The following question is discussed: which locally compact topological groups have an effective distal action on some compact metrizable space?

Let be a C*-algebra and τ:G → Aut a compact abelian action such that the fixed point algebra τ is simple. Denote by F the *-subalgebra of G-finite elements. Let H: F → be a *-operator commuting with τ such that and the matrix inequality

holds for all finite sequences X1, …, Xn in F. Then H is closable, and the closure is the generator of a strongly continuous semigroup {exp (−t): t ≥ 0} of completely positive contractions. Furthermore, there exists a convolution semigroup {μt: t ≥ 0} of probability measures on G such that

.

This result has various extensions and refinements.

Let G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

We show that there is a residual subset of the set of C1 diffeomorphisms on any compact manifold at which the map

is continuous. As this number is apt to be infinite, we prove a localized version, which allows one to conclude that if f is in this residual set and X is an isolated chain component for f, then

(i) there is a neighbourhood U of X which isolates it from the rest of the chain recurrent set of f, and

(ii) all g sufficiently C1 close to f have precisely one chain component in U, and these chain components approach X as g approaches f.

(ii) is interpreted as a generic non-bifurcation result for this type of invariant set.

We prove that on closed surfaces of higher genus cohomological invariants are not sufficient to distinguish topologically transitive flows which are not topologically conjugate; this contradicts a conjecture of Katok.

We shall measure how thick a basic set of a C1 axiom A diffeomorphism of a surface is by the Hausdorff dimension of its intersection with an unstable manifold. This depends continuously on the diffeomorphism. Generically a C2 diffeomorphism has attractors whose Hausdorff dimension is not approximated by the dimension of its ergodic measures.

An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C1 diffeomorphisms to C1 Hamiltonian vector fields.

This paper deals with the question of which periods can occur as periods of periodic points of zero entropy surface homeomorphisms in a given isotopy class. We give new examples of isotopy classes for which there are non-trivial restrictions and describe how the possible periods can be computed. Certain phenomena occur only for surfaces of large genus. These results have applications to the periodic data question for Morse–Smale maps.