Renormalizable correspondences and Hausdorff dimension

• Grant number: 16/16012-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: November 01, 2016
• End date: October 31, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Edson de Faria
• Grantee: Carlos Alberto Siqueira Lima
• Host 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)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

SIQUEIRA, CARLOS. HAUSDORFF DIMENSION OF JULIA SETS OF UNICRITICAL CORRESPONDENCES. Proceedings of the American Mathematical Society, v. N/A, p. 13-pg., 2022-10-28. (16/16012-6)

SIQUEIRA, CARLOS. Dynamics of hyperbolic correspondences. Ergodic Theory and Dynamical Systems, v. 42, n. 8, p. 32-pg., 2021-05-04. (16/16012-6)