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Renormalizable correspondences and Hausdorff dimension

Grant number: 16/16012-6
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): November 01, 2016
Effective date (End): October 31, 2017
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Edson de Faria
Grantee:Carlos Alberto Siqueira Lima
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Associated research grant:11/16265-8 - Low dimensional dynamics, AP.TEM

Abstract

We shall study the concept of renormalisation for a family of unicritical correspondences with rational exponents. Lyubich and Shishikura proved (independently) that if a quadratic map has no indifferent cycle and is only finite renormalizable, then its Julia set has measure zero. For non-integer exponents, Siqueira recently proved that the Julia set has Hausdorff dimension d< 2 for parameters sufficiently close to the origin. In this project we hope to link these two concepts, generalising Ruelle's formula for the solenoidal extension of the Julia set in C^2. (AU)