Differentiability of thermodynamic quantities for partially hyperbolic systems

• Grant number: 17/08732-1
• Support Opportunities: Regular Research Grants
• Duration: August 01, 2017 - July 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Suzete Maria Silva Afonso
• Grantee: Suzete Maria Silva Afonso
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil
• Associated researchers: Jaqueline Siqueira Rocha ; Vanessa Ribeiro Ramos

Abstract

In this project we propose the study of the thermodynamical formalism for a class of partially hyperbolic systems associated to regular potentials. We start with the study of a family of partially hyperbolic horseshoes for which we obtain linear response formulas for the equilibrium states associated to Hölder continuous potentials with small variation. Then we study partially hyperbolic skew-products whose base is a non-uniformly expanding map. For these systems we enlarge the class of potential for which we guarantee existence and finiteness of equilibrium states. Next we show that certain thermodynamical quantities such as topological entropy, topological pressure and correlation function (for the maximal entropy measure) are differentiable. Finally we consider step skew products, or skew products given by a iterated functions system and we exhibit sufficient conditions (on the iterated functions system and on the potentials) for the existence and finiteness of equilibrium states. (AU)