Patterns in random fractals
We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relat...
| Autores: | , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2020 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/168985 |
| Acesso em linha: | http://hdl.handle.net/11336/168985 |
| Access Level: | acceso abierto |
| Palavra-chave: | PATTERNS RANDOM FRACTALS FRACTAL PERCOLATION PROGRESSIONS https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer´edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies. |
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