Since the theory developed by Georg Cantor, mathematicians have taken a sharp interest in the sizes of infinite sets. We know that the set of integers is infinitely countable and that its cardinality is א0. Cantor proved in 1891 with the diagonal argument that the set of real numbers is uncountable and that there cannot be any bijection between integers and real numbers. Cantor states in partic...

By a topological dynamical system (Y, ψ), we mean a compact Hausdorff space Y endowed with a homeomorphism ψ. When (Yi, ψi), i = 0, 1 are two topological dynamical systems, ψ0 × ψ1-invariant closed subsets of Y0 × Y1 are called (topological) joinings, and when (Y0, ψ0) equals (Y1, ψ1), they are called self-joinings. In the measure-theoretical setting, the notion of selfjoinings was introduced b...

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.

Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of poin...

We investigate some product structures in R. Thompson’s group V , primarily by studying the topological dynamics associated with V ’s action on the Cantor set C. We draw attention to the class D(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D(V,C) contains all finite groups, the free group on two gen...

We suggest a Schauder basis in Banach spaces of smooth functions and traces of smooth functions on Cantor-type sets. In the construction, local Taylor expansions of functions are used. c ⃝ 2011 Elsevier Inc. All rights reserved.

We construct a large class of dynamically defined Cantor sets on the real line whose self-difference sets are Cantor sets of arbitrary positive measure. This relates to a question posed by J. Palis which arises naturally in the context of homoclinic bifurcations in dimension 2. §

The compact Hausdorff space X has the CSWP iff every subalgebra of C(X,C) which separates points and contains the constant functions is dense in C(X,C). Results of W. Rudin (1956), and Hoffman and Singer (1960), show that all scattered X have the CSWP and many non-scattered X fail the CSWP, but it was left open whether having the CSWP is just equivalent to being scattered. Here, we prove some g...

Equilibrium Cantor-type sets are suggested. This allows to obtain Green functions with various moduli of continuity and compact sets with preassigned growth of Markov’s factors.

In the operator version of the Hahn-Banach-Kantorovich theorem, the range space Y is assumed to be Dedekind complete. Y. A. Abramovich and A. W. Wickstead improved this by assuming only the Cantor property on Y . Along the same line of reasoning, we obtained in this paper several new results of this type. We also see that assuming Cantor property on the domain spaces instead gives good results,...