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Publication . Article . Preprint . 2016

Absolute continuity of self-similar measures, their projections and convolutions

Pablo Shmerkin; Boris Solomyak;
Open Access
English
Published: 01 Jul 2016
Publisher: American Mathematical Society
Country: Argentina
Abstract
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 one 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.
Comment: 33 pages, no figures
Subjects by Vocabulary

Microsoft Academic Graph classification: Bernoulli's principle Pure mathematics Convolution Parameter space Dimension (vector space) Measure (mathematics) Mathematics Projection (mathematics) Absolute continuity Hausdorff dimension

Subjects

ABSOLUTE CONTINUITY, SELF-SIMILAR MEASURES, HAUSDORFF DIMENSION, CONVOLUTIONS, Matemática Pura, Matemáticas, CIENCIAS NATURALES Y EXACTAS, //purl.org/becyt/ford/1.1 [https], //purl.org/becyt/ford/1 [https], Applied Mathematics, General Mathematics, Dynamical Systems (math.DS), Classical Analysis and ODEs (math.CA), FOS: Mathematics, Primary 28A78, 28A80, secondary 37A45, 42A38, Mathematics - Dynamical Systems, Mathematics - Classical Analysis and ODEs

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Funded by
NSF| Ergodic Theory, Dynamics and Fractals
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 0968879
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
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