By analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X, T) according to the various ways a sequence Tni can tend to the identity transformation. The main results obtained are:

(i) On a rigid flow there exists a T-invariant, symmetric, closed relation Ñ such that (X, T) is uniformly rigid iff Ñ = Δ, the diagonal relation.

(ii) For syndetically distal (hence distal) flows rigidity is equivalent to uniform rigidity.

(iii) We construct a family of rigid flows which includes Körner's example, in which Ñ exhibits various kinds of behaviour, e.g. Ñ need not be an equivalence relation.

(iv) The structure of flows in the above mentioned family is investigated. It is shown that these flows are almost automorphic.

We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).

So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?

Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

On étudie ici les homomorphismes métriques entre systèmes dynamiques definis par des substitutions de même longueur. En général, ces homomorphismes sont continus, et ont une forme simple. Le commutant essentiel d'un système dynamique defini par une substitution est fini, et peut être déterminé explicitement.

Let f be an orientation preserving ℋ-diffeomorphism of the circle. If the rotation number α = ρ(f) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation Rα. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h−lRαh. The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H1], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df ∈ Lp, p > 1, α = ρ (f) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of Rα, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that f is close to Rα; the proof is very different from Herman's and is very much in the spirit of [KO].

We give an algorithm for deciding the existence of epimorphisms of finitely additively generated Z[t] modules and dimension groups. This shows that the existence of eventual right resolving shift factor maps for mixing finite type shifts is decidable.

Let T be a translation defined on a nil-manifold N/Γ, where N is a nilpotent group of order two.

It is a general problem to find appropriate sets of moduli for families of functions that generate dynamical systems. In this paper we solve this problem for a specific family of holomorphic self-maps of ℂ* defined by

The main theorem states that any function topologically conjugate to a member of ℱ is holomorphically conjugate to some member of the family. It follows that the coefficients of the polynomials P(z) and Q(z) are a suitable set of moduli for the families of dynamical systems generated by these functions.

The moduli spaces of functions in ℱ are easy to study computationally and have been studied by many authors. (See references in the text.)

We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.

Given a manifold equipped with a free, properly discontinuous, cocompact ℤ-action, and a flow on which is ℤ-equivariant, we study the qualitative dynamics of . Under certain hypotheses on , we show that the chain recurrent set of has a decomposition which is the analogue, in the category of ℤ-equivariant flows, of Smale's spectral decomposition for recurrent sets of Axiom A flows.

Suppose we are given an action α: G → Aut (M) of a group G on a factor M; α possible way to analyse α may be to look at the invariant components where the action becomes more tractable. This point of view naturally leads to the study of the injective invariant subalgebras (recall for instance the good properties shared by amenable discrete or compact actions in the hyperfinite case [14]).

It is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

A class of percolation models on ℤd is developed in which the stochastic structure is provided by means of a d-parameter dynamical system. Of particular interest are those models generated by circle rotations. Unlike for independent models, the critical value and the percolation function can be explicitly calculated. These calculations lead to a conjecture concerning the behaviour of a related dynamical system.

We consider inner functions on the unit disk which have a finite number of singularities on the unit circle. The restriction of such functions to the circle are maps onto the circle. We give sufficient conditions that these restrictions are exact endomorphisms whose natural extensions are Bernoulli and that the entropy is given by Rohlin's formula, We also give the entropy in closed form if ƒ' is in the Nevalinna class N. An example is considered. In the last section we show that if two restrictions are metrically isomorphic, they are diffeomorphic.

It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.

We prove that an arbitrary one dimensional smooth dynamical system with non-degenerate critical points has no wandering intervals.

It is shown that a result by Moeckel holds not only for admissible subgroups of SL (2, ℤ), but also for arbitrary subgroups of finite index.

Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.

The notion of minimal self-joinings for conservative nonsingular actions is defined as a restriction on the nature of rational self-joinings. The need to consider rational joinings is demonstrated by showing that any two type II1 actions whose Cartesian product is ergodic have type IIIλ nonsingular joinings. Lastly, actions of all Krieger types with minimal self-joinings are constructed. Hence these actions are prime and commute only with their powers.

Letf: M → M be a C1 map on a compact manifold. We give a topological condition under which f has an even number of periodic points with a given period. We also obtain a sufficient condition, in terms of homology, for ƒ to have infinitely many periodic points.

In this paper we study some conjugacy invariants (moduli) for discrete two dimensional dynamical systems, with a homoclinic tangency. We show that the modulus obtained by Palis in the heteroclinic case also turns up in the case considered here. We also present two new conjugacy invariants.