Algorithm for determining pure pointedness of self-affine tilings

Overlap coincidence in a self-affine tiling in $\R^d$ is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in $\R^d$ and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several examples. In the course the proof of the algorithm, we show a variant of the conjecture of Urba\'nski (Solomyak \cite{Solomyak:08}) on the Hausdorff dimension of the boundaries of fractal tiles.Comment: 21 pages, 3 figure

Generación de curvas fractales a partir de homomorfismos entre lenguajes [con Mathematica®]

In this paper we implement with the software Mathematica 8.0 some combinatorial properties of Fibonacci Word, which can be generated from the iteration of a homomorphism between languages.We collect also some graphic properties of the fractal curve associated to this word, which can be generated from drawing rules similar to those used in the L-Systems. All codes used in this paper are presented in detail and then they are applied to generate new fractal curves. We conclude with an alternative way to generate the Fibonacci curve and other curves from characteristics words.En este artículo se hace una implementación con el software Mathematica 8.0 de algunas propiedades combinatorias de la cadena o palabra de Fibonacci, la cual se puede generar a partir de la iteración de un homomorfismo entre lenguajes. Asimismo se recopilan algunas propiedades gráficas de la curva fractal asociada a esta cadena de símbolos, la cual se puede generar a partir de unas reglas de dibujo análogas a las utilizadas en los L-Sistemas. Todos los códigos utilizados en el artículo se presentan en detalle y luego se aplican para generar nuevas curvas fractales. Finalizamos con una forma alternativa de generar la curva de Fibonacci y otras curvas a partir de cadenas características.   &nbsp

Self-Affine Tiles in Rn

AbstractAself-affine tilein Rnis a setTof positive measure withA(T)=∪d∈D(T+d), whereAis an expandingn×nreal matrix with |det(A)|=man integer, and D={d, d2, ..., dm}⊆Rnis a set ofmdigits. It is known that self-affine tiles always give tilings of Rnby translation. This paper extends known characterizations of digit sets D yielding self-affine tiles. It proves several results about the structure of tilings of Rnpossible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings