Assouad dimension and local structure of self-similar sets with overlaps in Rd

• Autores: Garcia, Ignacio Andres
• Año de publicación: 2020
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Assouad dimension; Self-similar sets; Overlaps
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

For a self-similar set in Rd that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least 1. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of overlapping directions and the Assouad dimension of the orthogonal projection of the self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on the orthogonal parts of the similarities so that the self-similar set has Assouad dimension bigger than 2, and also to answer a question posed by Farkas and Fraser. The result is also extended to the context of graph directed self-similar sets. The proof of the result relies on finding an appropriate weak tangent to the set. This tangent is used to describe partially the topological structure of self-similar sets which are both attractors of an iterated function system not satisfying the weak separation property and of an iterated functions system satisfying the open set condition.
Fil: Garcia, Ignacio Andres. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Físicas de Mar del Plata. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Físicas de Mar del Plata; Argentina