Absolute continuity of self-similar measures, their projections and convolutions
We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research conce...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2016 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/90812 |
| Acceso en línea: | http://hdl.handle.net/11336/90812 |
| Access Level: | acceso abierto |
| Palabra clave: | ABSOLUTE CONTINUITY SELF-SIMILAR MEASURES HAUSDORFF DIMENSION CONVOLUTIONS https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets. |
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