Updated: Oct 26, 2019
Earlier, I made two posts talking about the boundary at infinity, and why it's a great way to talk about sets of geodesics in hyperbolic 2-space. Now, that we know how to talk about sets of geodesics, there's a reasonable question that follows: How do we talk about measuring sets of geodesics?
A certain type of measure on the space of geodesics in the universal cover is called a geodesic current. Before talking about the formal definition, let's think about what kinds of properties it would make sense for something like this to have.
We want it to behave well in a general sense, so let's say we want it to be Borel.
If A and B project to the same geodesics on the surface (well, its unit tangent bundle), then we might want them to have the same measure. So let's choose measures that are invariant under deck transformations.
I always want to just look at probability measures if I can, but let's see if having a finite measure makes sense here. Suppose you have a set of geodesics, A, and it's measure is n. Deck transformations are properly discontinuous, and so there are infinitely many deck transformations of A that are pairwise disjoint. In 2, we also asked that this measure is invariant under deck transformations. Measures are countably additive (and the fundamental group is countable), so the measure of all pairwise disjoint deck transformations of A is an infinite sum over the measure of A. This means the measure of all the geodesics must also be infinite. So, if we want any set of geodesics to have positive measure and to be deck transformation invariant, it's going to be an infinite measure. We can ask for something reasonable though, and have these measures be locally finite. That is, we can always take a set to be small enough that it will have finite measure.
These are good things to ask for! For the space of geodesics G(g) on the universal cover of a negatively curved surface S with metric g, let's define a nice type of measure.
Definition: A geodesic current is a Borel, locally finite, deck transformation invariant measure on G(g).
Let's remember what a set in G(g) might look like. We can define a set A by looking at two sets on the boundary and considering all the geodesics with one endpoint in each set.
Some approximation of that picture is here. I'm thinking about unoriented geodesics, so it doesn't matter which end is which.
I'm going to give two examples of geodesic currents. Both of them show up in the papers that I'm thinking about right now. In fact, I haven't actually run into any other examples, and I can explain why that is a little later.
Closed orbit currents
These currents have, you guessed it, something to do with closed orbits. Choose a nontrivial homotopy class of your surface and then choose it's geodesic representative. A nontrivial class here means that the curves in the class can't contract to a point. We're in negative curvature, so this representative is unique. If you're more comfortable with "conjugacy class" or "class of the fundamental group" than "homotopy class", those are fine as well! Then, look at lifts of that geodesic in the universal cover.
Definition: The closed orbit current based at a class is
That is, this measure literally counts how many lifts of that geodesic representative are in the set your feed to it. For example, take the set A to be all the geodesics passing between the blue areas on the boundary, and take the red curves to be all the lifts of the geodesic gamma that we care about. (There are infinitely many, but that picture wouldn't be very helpful). Since only one of those red geodesics has an endpoint in each of the blue regions, then this closed orbit current only counts that one. By that I mean that
I think that this geodesic current is a very nice one to work with, and not at all scary like I expected geodesic currents to be. Next time I'll define another geodesic current. At some time in the future, I'll tell you what nice geometric information we can get by pairing geodesic currents!