Effective mapping class group dynamics II: Geometric intersection numbers

Abstract

We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the boundary at infinity of Teichmüller space, an effective estimate comparing the Teichmüller and Thurston metrics along mapping class group orbits of Teichmüller space, and, in the sequel, effective estimates for countings of filling closed geodesics on closed, negatively curved surfaces.

Effective mapping class group dynamics III: Counting filling closed curves on surfaces

“…Outline of this section. In this section we discuss the main results of the prequels [Ara20b] and [Ara21] needed to apply the tracking method sketched in §1 to prove Theorem 1.1. We first discuss the effective estimates for counting functions of mapping class group orbits of Teichmüller space proved in the prequel [Ara20b].…”

“…We refer to this method as the tracking method. This method relies on recent progress made in the prequels [Ara20b] and [Ara21] on the study of the effective dynamics of the mapping class group on Teichmüller space and the space of closed curves of a closed, orientable surface. These recent developments in turn rely on the exponential mixing rate, the hyperbolicity, and the renormalization dynamics of the Teichmüller geodesic flow as their main driving forces.…”

“…We refer to this method as the tracking method. At the heart of this method is the tracking principle proved in the prequel [Ara21]. According to this principle, the action of the mapping class group on the space of closed curves of a closed, orientable surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents.…”

“…In §2 we cover the background material and set up the notation that will be used throughout the rest of this paper. In §3 we discuss the main results of the prequels [Ara20b] and [Ara21] concerning the effective dynamics of the mapping class group on Teichmüller space and the space of closed curves of a closed, orientable surface. In §4 we study the regularity of geometric intersection numbers and extremal lengths on the space of singular measured foliations of a closed, orientable surface.…”

Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves

“…Even more recently, in [Ara21b], novel methods were introduced by the author to prove analogous effective estimates for countings of filling closed geodesics of a given topological type. These methods rely on recent progress made in the prequels [Ara20] and [Ara21a] on the study of the effective dynamics of the mapping class group on Teichmüller space and the space of closed curves of a closed, orientable surface. These recent developments in turn rely on the exponential mixing rate, the hyperbolicity, and the renormalization dynamics of the Teichmüller geodesic flow as their main driving forces.…”