We study the space H(d) of continuous Z-actions on the Cantor set, particularly questions on the existence and nature of actions whose isomorphism class is dense (Rohlin’s property). Kechris and Rosendal showed that for d = 1 there is an action on the Cantor set whose isomorphism class is residual; we prove in contrast that for d ≥ 2 every isomorphism class in H(d) is meager. On the other hand,...

1 It is known that not every Cantor set of S is C-minimal. In this work we prove that every member of a subfamily of the called regular interval Cantor set is not C-minimal. We also prove in general, for a even large class of Cantor sets, that any member of such family can be C-minimal, for any 2 > 0.

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger than three. These Cantor sets in S are constructed by using Bing or Whitehea...

In this paper, we present a generalized form of the Cantor ternary set by studying cantor middle where 1 and its fractal dimension. The paper also introduces Heine-Borel shows that generalised are sets.

We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theo...

We deene the n-lift of a one-dimensional system x i+1 = f (x i). The n-lift can be thought of as a perturbation of the one-dimensional system depending on the state of the system n ? 1-time steps back. We prove that certain f-invariant Cantor sets give invariant Cantor sets in the lifted system. We prove that if f has an invariant hyperbolic Cantor set then the lifted system has an invariant hy...

A technique based on Cantor division is used to embellish artistic knotwork designs. Weaving rules are relaxed to allow a mostly alternating weave to be applied to the resulting patterns. The new method is compared to traditional knotwork, and the generation of examples is discussed. r 2005 Elsevier Ltd. All rights reserved.

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ω of irrationals, or certain of its subspaces. In particular, given f ∈ (ω\{0}), we consider compact sets of the form Q i∈ω Bi, where |Bi| = f(i) for all, or for infinitely many, i. We also consider “n-splitting” compact sets, i.e., compact sets K such that for any f ∈ K and i ∈ ω, |{g(i) : ...

The Idea for this article was given by a problem in real analysis. We wanted to determine the one-dimensional Lebesgue-measure of the set f(C)9 where C stands for the classical triadic Cantor set and/is the Cantor-function, which is also known as "devil's staircase." We could see immediately that to determine the above measure we needed to know which dyadic rationals were contained in C. We soo...

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor set...