Logarithm laws for equilibrium states in negative curvature

Frédéric Paulin (1) , Mark Pollicott (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Department of Mathematics, University of Warwick

Let $M$ be a pinched negatively curved Riemannian manifold, whose unit tangent bundle is endowed with a Gibbs measure $m_F$ associated to a potential $F$. We compute the Hausdorff dimension of the conditional measures of $m_F$. We study the $m_F$-almost sure asymptotic penetration behaviour of locally geodesic lines of $M$ into small neighbourhoods of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objects. As an arithmetic consequence, we give almost sure Diophantine approximation results of real numbers by quadratic irrationals with respect to general Hölder quasi-invariant measures.

Mots clés: Gibbs measure; pressure; Hausdorff dimension; Diophantine approximation; quadratic irrational; geodesic flow; negative curvature; spiraling; Khintchine theorem; logarithm law; equilibrium states

Domaines: Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]; Géométrie différentielle [math.DG]; Théorie des nombres [math.NT]

Frédéric Paulin, Mark Pollicott. Logarithm laws for equilibrium states in negative curvature. 2014. ⟨hal-00988681⟩