If we add the well-known fact that any subset N of the Cantor set C homeomorphic to the irrationals is never Fσ, we have given all the reasons why the pair (C, N) is called the Hurewicz test for Fσ sets. Since the work of Hurewicz Theorem 1 has been strengthened in many successive steps. We will discuss this in the next section. Before doing this we recall some basic definitions in descriptive ...

The invariant Cantor sets of the logistic map gμ(x) = μx(1 − x) for μ > 4 are hyperbolic and form a continuous family. We show that this family can be obtained explicitly through solutions of infinitely coupled differential equations due to the hyperbolicity. The same result also applies to the tent map Ta(x) = a(1/2− |1/2− x|) for a > 2.

We show that there is a computable Boolean algebra B and a computably enu-merable ideal I of B such that the quotient algebra B=I is of Cantor{Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of innnite Cantor{Bendixson rank.

We calculate all ergodic measures for a specific function F on the unit interval. The supports of these measures consist of periodic orbits of period 2n and the classical ternary Cantor set. On the Cantor set, F is topologically conjugate to an “adding machine” in base 2. We show that F is representative of the class of functions with zero topological entropy on the unit interval, already analy...

We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B=I is of Cantor{Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of in nite Cantor{Bendixson rank.

We study the degree of fractality for the electromagnetic field diffracted from gratings and zone plates with internal Cantor structure. This fractality is determined by the correlation peaks which appear in the self-similarity function for the intensity distribution of the field. The results are complementary to the study of the homogeneity property, presented in a previous work [7]. Key-Words...

The one purpose of this little Note is to show that formal arguments need not be lengthy at all; on the contrary, they are often the most compact rendering of the argument. Its other purpose is to show the strong heuristic guidance that is available to us when we design such calculational proofs in sufficiently small, explicit steps. We illustrate our approach on Georg Cantor’s classic diagonal...

In this paper we analyze Cantor type sets constructed by the removal of open intervals whose lengths are the terms of the p-sequence, {k−p}∞k=1. We prove that these Cantor sets are s-sets, by providing sharp estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal points of the boundaries of the so-called random stable zonotopes.

RÉSUMÉ. Deux instruments de synthèse vocale sont présentés : le Cantor Digitalis et le Digitartic. Ces deux instruments utilisent des gestes bimanuels dérivés de l’écriture ou du dessin sur une tablette graphique. Le signal est calculé par un synthétiseur vocal paramétrique, comprenant des modèles de source voisée et de bruits consonantiques et une structure à formants série/parallèle, pour le ...

We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A∩C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.