Complex dynamics and thermodynamic formalism

Author(s)

Carlos Alberto Siqueira Lima

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2011-04-01

Examining board members

Daniel Smania Brandão; Katrin Grit Gelfert; Artur Oscar Lopes

• Advisor: Daniel Smania Brandão

Abstract

We shall study complex dynamical systems in the Riemann sphere and prove that the Hausdorff dimension \'dim IND. H\' J( \'f IND. Lãmbda\' ) of the Julia set J( \'f IND. lâmbda\' ) of an holomorphic family of hyperbolic rational maps \'f IND. lâmbda\' defines a real analytic map of the parameter \'lâmbda\': This result was proved in 1981 by D. Ruelle (see [44]). We give an alternative proof using holomorphic motions (see [31]), which was originally developed to study the structural stability problem of complex dynamical systems. Throughout this work, we shall use several tools of Thermodynamic Formalism, including Bowens formula (AU)

• FAPESP's process: 08/07356-7 - Complex Dynamics and Thermodynamic Formalism
• Grantee: Carlos Alberto Siqueira Lima
• Support Opportunities: Scholarships in Brazil - Master