Geodesic Currents and Counting Problems

Abstract

For every positive, continuous and homogeneous function f on the space of currents on a compact surface Σ, and for every compactly supported filling current α, we compute as L → ∞, the number of mapping classes φ so that f (φ(α)) ≤ L. As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichmüller space equipped with the Thurston metric.

From curves to currents

“…First note that the set is compact because f is positive on nonzero measured laminations. Let ; then , as proved by Rafi and Souto [51, p. 879]. Finally, we apply Mirzakhani’s counting result [43, Theorem 1.3] and the Portmanteau theorem (see [6, Theorem 30.12]) to conclude the limit exists.…”

“…Thus, geodesic currents allow one to treat curves and metric structures on surfaces as the same type of object. Via this unifying framework, counting curves of a given topological type and counting lattice points in the space of deformations of geometric structures become the same problem [51, Main Theorem]. Geodesic currents also play a key step in the proof of rigidity of the marked length spectrum for metrics, via an argument by Otal [47, Théorème 2].…”

“…The problem of extending functions to geodesic currents is interesting in itself, since, by a result of Rafi and Souto reviewed in Section 5, it provides a way to compute asymptotics of the number of curves of a fixed type with a bounded ‘length’, for a notion of ‘length’ that extends to currents [51]. Their result builds on work by Mirzakhani [44, Theorem 7.1] and Erlandsson–Souto [24, Proposition 4.1].…”

On normalizations of Thurston measure on the space of measured laminations

“…We call these normalizations (rigorously defined in the Section 2) symplectic and integral Thurston measures on ML(Σ) and denote them by µ ω , µ Z , respectively. In this paper we answer the question raised in [RSo17]: What is the ratio between measures µ ω and µ Z ?…”

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

“…For every L > 0 consider the counting function cur( , α, L):= #{β ∈ Mod g • α | (β) ≤ L}.In[RS19], Rafi and Souto introduced novel methods for studying the asymptotics of these counting functions. Given any filling geodesic current α ∈ C g consider the finite positive constantc(α) := µ Thu ({λ ∈ ML g | i(α, λ) ≤ 1}) /#Stab(α).Theorem 6.3.…”