From curves to currents

Abstract

Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

Distribution in the unit tangent bundle of the geodesics of given type

“…The reason why we care about continuous, homogeneous and positive functions on is that there is such a function satisfying for every closed geodesic , where the first is the function we are talking about and where the second one is just the length of the geodesic . The reader can see [8, 14] for other examples of continuous homogeneous functions on the space of currents—in fact we will encounter yet other such functions below.…”

Counting curves on orbifolds

“…on the space of currents, where homogeneous means that 𝐹(𝑡 ⋅ 𝜆) = 𝑡 ⋅ 𝐹(𝜆). See [5,12] for many examples of such functions. Let us now describe the strategy of the proof of our main result, Theorem 1.2.…”

“…In fact, we can replace $$\ell _{\operatorname{O}}$$ by any continuous homogenous function $$\begin{equation*} F:\mathcal {C}^{\operatorname{or}}(\operatorname{O})\rightarrow \mathbb {R}_{\geqslant 0} \end{equation*}$$on the space of currents, where homogeneous means that $$F(t\cdot \lambda )=t\cdot F(\lambda )$$. See [5, 12] for many examples of such functions. Theorem Let $$\operatorname{O}$$ be a compact orientable hyperbolic orbifold with possibly empty totally geodesic boundary and let $$\mathcal {C}^{\operatorname{or}}(\operatorname{O})$$ be the associated space of geodesic currents.…”