Logarithm laws for equilibrium states in negative curvature

TL;DR

This paper investigates the statistical and geometric properties of geodesic flows in negatively curved manifolds, establishing logarithm laws and Hausdorff dimension results, with applications to Diophantine approximation.

Contribution

It provides new Hausdorff dimension calculations for Gibbs measures and proves logarithm laws for geodesic spiraling behavior in negatively curved spaces.

Findings

Hausdorff dimension of conditional Gibbs measures computed

Logarithm laws established for geodesic spiraling

Almost sure Diophantine approximation results derived

Abstract

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 H\"older quasi-invariant measures.