In this paper, the optical properties of one dimensional fractal structures are investigated. We consider six typical fractal photonic structures: the symmetric dual cantor-like fractal structure, the asymmetric dual cantor-like fractal structure, the single cantor-like fractal structure, the symmetric dual golden-section fractal structure, the asymmetric dual golden-section fractal structure a...

Cantor diffractals are waves that have encountered a Cantor grating. In this paper, we report an important property of Cantor diffractals, namely that of redundancy. We observe that the Fraunhofer diffraction pattern comprises of several bands, each containing complete information about the fractal aperture. This redundancy allows for a faithful reconstruction ...

A continuous open finite-to-one surjection f preserves Cantor rank. If the domain of f is a Hausdorff compact space, Cantor degree variations are bounded by the maximal size of the finite fibres. If the fibres are infinite, we show inequalities involving the maximal and minimal rank of the fibres. A closed preorder on a denumerable Hausdorff compact space is the intersection of clopen preoders....

In the year (1879-1884), George Cantor coined few problems and consequences in the field of set theory. One of them was the Cantor ternary set as a classical example of fractals. In this paper, 5-adic Cantor one-fifth set as an example of fractal string have been introduced. Moreover, the applications of 5-adic Cantor one-fifth set in string theory have also been studied.

Wave propagation through a subclass of deterministic nonperiodic media, namely, fractal Cantor multilayer structures are investigated theoretically as well as experimentally. Transmission spectra of Cantor structures are found to have two distinctive properties (scalability and sequential splitting) closely related to the geometrical peculiarities of the multilayers. A systematic correlation be...

We construct a nonarchimedean (or p-adic) analogue of the classical ternary Cantor set C. In particular, we show that this nonarchimedean Cantor set C3 is self-similar. Furthermore, we characterize C3 as the subset of 3-adic integers whose elements contain only 0’s and 2’s in their 3-adic expansions and prove that C3 is naturally homeomorphic to C. Finally, from the point of view of the theory ...

We define the ε-distortion complexity of a set as the shortest program, running on a universal Turing machine, which produces this set at the precision ε in the sense of Hausdorff distance. Then, we estimate the ε-distortion complexity of various central Cantor sets on the line generated by iterated function systems (IFS’s). In particular, the ε-distortion complexity of a C Cantor set depends, ...

Not every Cantor set can arise as the minimal set for a C 1 diieomor-phism of the circle. For example, D. McDuu has shown that the usual \middle thirds" Cantor set cannot arise this way. In this note we exclude a suitable neighborhood of the class of aane Cantor sets.

A Cantor system is defined. The geometry of a certain family of Cantor systems is studied. Such a family arises in dynamical systems as hyperbolicity is created. We prove that the bridge geometry of a Cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap.