Equilibrium states in negative curvature

Frédéric Paulin (1) , Mark Pollicott (2) , Barbara Schapira (3, 4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Department of Mathematics, University of Warwick
3 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
4 IRMAR - Institut de Recherche Mathématique de Rennes

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications to the variational principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on some Riemannian covers.

Mots clés: variational principle; strong unstable foliation; geodesic flow; negative curvature; strong unstable foliation.; Patterson density; pressure; orbit counting; periods; Gibbs state

Domaines: Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]

Frédéric Paulin, Mark Pollicott, Barbara Schapira. Equilibrium states in negative curvature. SMF, 373, pp.viii+281, 2015, Astérisque, 978-2-85629-818-3. ⟨hal-00758114⟩