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Análisis fractal de conjuntos de Cantor no lineales

Ignacio Andrés Garcia Ursula María Molter Pablo Panzone Tomás Godoy Carlos Gustavo Tamm de Araújo Moreira Roberto Aníbal Scotto

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We study Cantor sets of zero Lebesgue measure associated to non increasing sequences, which gives the size of complementary bounded intervals. Each of these sets has associated a dimension function h such that the h-dimensional Hausdorff measure is positive and finite. We give a characterization of the Hausdorff and packing measure of these sets, and from this we can decide when the function h is equivalent to some power function. We show that the C_p Cantor set, which is the associated to the sequence {1/n^p}, p>1, is the attractor of an iterated function system (IFS) with 1/p-Hölder continuous derivatives, being this the highest degree of smoothness that can reach any other system which has C_p as its attractor. It is also proved that this system is dynamically conjugated to the classic system of similitudes that has the 1/2^p-middle Cantor set as attractor. Finally we study the convolution measure n_p*n_q, where n_p is the invariant measure of the IFS associated to C_p with weights (1/2,1/2). This problem is related with the topological structure and dimension of the sum set C_p+C_q. We extend results on r-middle Cantor sets to obtain that the Hausdorff dimension of this set is the sum of the dimension of each set for almost everywhere p and q such that the sum of the dimensions does not exceed 1. On the other hand, for almost all cases when the sum of the dimensions exceed 1 we have that the convolution measure is absolutely continuous.
Palabras clave – provistas por el repositorio digital

Cantor sets; Dimension functions; Packing measure; Hausdorff dimension; Iterated function system; Convolution of measures; Conjuntos de Cantor; Funciones de dimensión; Medida packing; Dimensión de Haudorff; Sistema iterado de funciones; Convolución de medidas

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tesis

Idiomas de la publicación

  • español castellano

País de edición

Argentina

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