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 show how a nonlinear system that supports solitons can be driven to generate exact (regular) Cantor set fractals. As an example, we use numerical simulations to demonstrate the formation of Cantor set fractals by temporal optical solitons. This fractal formation occurs in a cascade of nonlinear optical fibers through the dynamical evolution from a single input soliton.

In this paper, we study the prototype of fractal of the classical Cantor middle-third set which consists of points along a line segment, and possesses a number of fascinating properties. We discuss the construction and the self-similarity of the Cantor set. We also generalized the construction of this set and find its fractal dimension.

Nazarov, Peres and Volberg proved recently in [8] that the Favard length of the n-th iteration of the four-corner Cantor set is bounded from above by n−c for an appropriate c. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.

The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional...

In a recently published paper (J. of Modern Optics 50 (9) (2003) 1477-1486) a qualitative analysis of the moiré effect observed by superposing two grids containing Cantor fractal structures was presented. It was shown that the moiré effect is sensible to variations in the order of growth, dimension and lacunarity of the Cantor fractal. It was also verified that self-similarity of the original f...

The purpose of this paper is to introduce the concept of fractals and its use in antenna arrays for obtaining multiband property. One type of fractals namely, Cantor set is investigated. Cantor set is used in linear array antenna design. Therefore, this array know fractal Cantor linear array antenna. A comparison with conventional non-fractal linear array antenna is made regarding the beamwidth...

This paper deals with strange attractors of S-unimodal maps f . It generalizes results from [BKNS] in the sense that very general topological conditions are given that either i) guarantee the existence of an absorbing Cantor set provided the critical point of f is sufficiently degenerate, or ii) prohibit the existence of an absorbing Cantor set altogether. As a byproduct we obtain very weak top...

Popular texts typically assert that guilds of craftsmen `̀ monopolized’ ’ markets in medieval England. Norman Cantor’s Medieval Reader declares `̀ craft guilds’ . . . main purpose and activity was narrow regulation of industrial productivity in order to restrain competition’ ’ (Cantor 1994, p. 278). Douglass North’s Structure and Change in Economic History asserts `̀ . . . guilds organized to prot...