Thermodynamic formalism for Lorenz maps

Renaud Leplaideur (1) , Vilton Pinheiro (2)

1 LM - Laboratoire de mathématiques de Brest
2 UFBA - instituo de matematica

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any Ho ̈lder continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.

Mots clés: Lorenz attractor; equilibrium states; unstable manifold; Markov set; non-uniformly hyperbolic dynamical system; induction scheme

Domaines: Systèmes dynamiques [math.DS]

Renaud Leplaideur, Vilton Pinheiro. Thermodynamic formalism for Lorenz maps. 2012. ⟨hal-00730551⟩