Notes for the summer school Symbolic dynamics and homeomorphisms of the Cantor set,

In this paper, we prove that for every integer k ≥ 1, the k-abelian complexity function of the Cantor sequence c = 101000101 · · · is a 3-regular sequence.

We construct a braided version BV of Thompson’s group V that surjects onto V . The group V is the third of three well known groups F , T and V created by Thompson in the 1960s that have been heavily studied since. See [6] and Section 4 of [5] for an introduction to Thompson’s groups. The group V is a subgroup of the homeomorphism group of the Cantor set C. It is generated by involutions [2, Sec...

We show that, for some Cantor sets in R, the capacity γs associated to the s-dimensional Riesz kernel x/|x| is comparable to the capacity Ċ 2 3 (d−s), 3 2 from non linear potential theory. It is an open problem to show that, when s is positive and non integer, they are comparable for all compact sets in R. We also discuss other open questions in the area.

The classical middle-thirds Cantor set can be described as follows. Start by taking C 0 = [0, 1], the unit interval in the real line. Then put which is to say that one removes the open middle third from the unit interval to get a union of two disjoint closed intervals of length 1/3. By repeating the process one gets for each nonnegative integer j a subset C j of the unit interval which is a uni...

Entrevista com José Guilherme Cantor Magnani

We discuss the domain-theoretic and topological content of the operator calculus used in the Irish School of the Vienna Development Method (VDM♣) of formal systems development. Thus, we examine the Scott continuity, or otherwise, of the basic operators used in this calculus when viewed as operators on the domain (X → Y ) of partial functions mapping X into Y . It turns out that the override, on...

We show that every minimal, free action of the group Z2 on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, Z-actions and Z2-actions.

We prove that a continuous function on the dual Cantor set is the scaling function of a uniformly symmetric circle endomorphism if and only if it satisfies a summation condition and compatibility condition. We use this result to establish an isomorphism between the space of continuous functions on the dual Cantor set satisfying these conditions and a Teichmüller space.

A distortion theory is developed for S−unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S−unimodal maps is classified according to a distortion property, called the Markov-property.