We propose a theory of chaos for discrete systems, based on their representation in a space of " binary histories " , B ∞. We show that B ∞ is a metrizable Cantor set which embeds the attractor Λ, itself also a Cantor set.ciation Générale pour la Coopération et le Développement (Belgium).

Many examples exist of one-dimensional systems that are topologically conjugate to the shift operator on Σ2 and are thus chaotic. Most of these examples which have invariant Cantor subsets, have Cantor subsets of measure zero. In this paper we outline the formulation of a C map on a closed interval that has an invariant Cantor subset of positive Lebesgue measure. We also survey techniques used ...

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (Gn) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson ωn.

This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width. 2000 Mathematics Subject Classification: 54A25, 54A35, 54G12,03E05, 06E15.

Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product C∗-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via K-theory and C∗-algebras. For example, it is shown that two Cantor minimal systems are approximately τ -conjugate if and o...

In this paper we explore the ambient quasiconformal homogeneity of planar domains and their boundaries. We show that the quasiconformal homogeneity of a domain D and its boundary E implies that the pair (D,E) is in fact quasiconformally bi-homogeneous. We also give a geometric and topological characterization of the quasiconformal homogeneity of D or E under the assumption that E is a Cantor se...

Given a real sequence (xn), we examine the set of all sums of the form P i∈I xi, as I varies over subsets of the natural numbers. We call this the achievement set of (xn), and write it AS(xn). For instance, AS(1/2) = [0, 1] by the existence of binary expansions, and AS(2/3) is the Cantor middle third set. We explore the properties of these two sequences that account for their very different ach...

We develop a machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: (1) The product of meager/null-additive sets in the Cantor space is meager/nulladditive, respectively. (2) The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measur...

We consider infinite dimensional Hamiltonian systems. First we prove the existence of “Cantor manifolds” of elliptic tori -of any finite higher dimensionaccumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are “branching” points of other Cantor manifolds of higher dimensional tori. We also provide a ...

The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativisation of the Cantor topology, the Uδ -topology introduced in...