Existência e Unicidade de Medidas de Equilíbrio para o Fluxo Geométrico de Lorenz

Abstract

This thesis addresses the problem of uniqueness of equilibrium states for the geometric Lorenz flow within the framework of thermodynamic formalism. The system is analyzed through its representation as a suspension flow over a two-dimensional Poincaré map P(x,y) = (L(x), g(x,y)), which is a partially hyperbolic extension of a one-dimensional Lorenz-like map L. While previous results by Bronzi and Oler [BO18] established generic uniqueness for the one-dimensional case, the transition to higher dimensions and continuous time remained a technical challenge due to transverse contraction and the flow's singularity.

Our main result proves that, for the space of piecewise Hölder potentials, there exists an open and dense subset H for which the flow admits a unique equilibrium state. The proof is based on a hierarchical approach that begins with the geometric description of the attractor and the formalization of dimensional correspondences (Chapter 1). We then proceed to demonstrate generic uniqueness for the two-dimensional base map (Chapter 3) and the development of a shifted operator J which, through Abramov's formulas, allows for the transfer of openness and density properties from the base to the suspension flow (Chapter 4). This work unifies and extends existing literature, consolidating the genericity of uniqueness in singular-hyperbolic systems.