---
title: The horocycle flow
lead: Fold a horizontal line of the hyperbolic plane onto the modular surface and it closes into a loop of length 1/y; as the line sinks the loop spreads evenly over the whole surface, at a pace set by the zeros of zeta.
prerequisites: ford-circles, riemann-zeta-function
---

Take the points above the real line, `z = x + i y` with `y > 0`, and measure them with a ruler that shrinks as you go down: a step of ordinary length `s` at height `y` counts as `s/y`. That is the upper half plane, the simplest home of hyperbolic geometry. The real line is now infinitely far away, and area is counted the same way, a small patch of ordinary area `A` at height `y` counting as `A/y^2`.

The moves of this geometry include the maps `z -> (a z + b)/(c z + d)` with `a`, `b`, `c`, `d` whole numbers and `a d - b c = 1`. The matrices form the group `SL(2, Z)`. Each one keeps the half plane, keeps the ruler, and sends circles and lines to circles and lines. Two of them are enough to build all the rest: the shift `z -> z + 1` and the flip `z -> -1/z`.

Now declare two points the same whenever one of these moves carries one to the other. Every point can be moved into the doorway-shaped region where `abs(x) <= 1/2` and `abs(z) >= 1`, and only points on its edges can be moved again: the shift glues the left wall to the right wall, and the flip folds the floor arc onto itself. What is left is the modular surface. Its top runs up for ever into a thin horn, the cusp, yet its total area is finite, exactly `pi/3`.

A horocycle is what the horizontal line `Im z = y` is in this geometry. It is not straight, since the straight lines here are the vertical lines and the half circles standing on the real line; it is a circle whose centre has gone off to infinity. The moves carry it to ordinary circles resting on the real line: the move whose left column is `a`, `c` sends it to the circle tangent at the fraction `a/c` with diameter `1/(c^2 y)`. At height 1 these are exactly [the Ford circles](/wiki/ford-circles/), diameter `1/c^2` over each fraction `a/c`, so the whole Ford picture is one horocycle seen many times over.

Because the shift slides the line along itself, on the surface the line closes up into a loop, the stretch from `x = 0` to `x = 1`. Its length is `1/y`, one unit of ordinary length at height `y`. The horocycle flow slides every point along its horocycle at unit speed (strictly it moves a point together with a direction, but the picture is the same), and these loops are its closed orbits: a point on the loop at height `y` comes back to itself after time `1/y`.

The figure is the loop at height `1/100`, of length 100, folded into the doorway and cut off at height about 2.47, which hides the top of the cusp and about 39% of the loop's length. Each point `x + i/100` is moved into the region, and a whole stretch of the line comes along with it as one arc, a piece of the circle tangent at `a/c` with diameter `100/c^2`. The arcs are tinted from blue for `c = 1`, the nearly straight ones, to pink for `c = 10`, the small caps along the floor. They crowd toward the floor because the loop spreads by area, and a patch low down counts for more area than the same patch high up.

Each arc comes from a fraction. Walk along the line and it breaks into stretches, one above each fraction between 0 and 1 whose bottom number `c` is at most about `1/sqrt(y)`. Above a fraction with bottom number `c` a move lifts the line as high as `1/(c^2 y)`, and the gluing of the walls cuts that stretch into one or more arcs, pieces of copies of one circle shifted sideways by whole numbers. So the folding is run by [the Farey sequence](/wiki/farey-sequence/) of order about `1/sqrt(y)`, and each bottom number `c` owns `phi(c)` stretches, one per fraction.

As the line sinks, `y -> 0`, the loop grows longer and does not bunch up anywhere: the share of its length inside any region of the surface tends to that region's share of the area `pi/3`. Put another way, for any smooth test function `f` on the surface that vanishes high in the cusp, the average of `f` along the loop tends to the average of `f` over the whole surface. The loop equidistributes. This is the theorem of [Zagier 1981](https://doi.org/10.1007/978-3-662-00734-1_10) for the modular surface, and of [Sarnak 1981](https://doi.org/10.1002/cpa.3160340602) for every hyperbolic surface of finite area with a cusp.

How fast it happens is where zeta comes in. The tool is the Eisenstein series, a function on the surface built by adding `Im(g z)^s` over the moves `g`, one for each bottom row `c`, `d` with no common factor, up to sign, since the value depends only on that row. Blend the loop averages at every height into one function of `s` and you get exactly `f` weighed against that series. The series carries `zeta(2 s)` in a denominator, so every zero `rho` of [the Riemann zeta function](/wiki/riemann-zeta-function/) in its strip, `0 < Re(rho) < 1`, becomes a pole at `s = rho/2` and leaves a wave in the error of size about `y^(1 - Re(rho)/2)`.

These zeros all have real part below 1, and that makes the error smaller than `y^(1/2)`, a statement as strong as the prime number theorem. These zeros all sit on the line of real part `1/2` exactly when the error stays below a constant times `y^(3/4 - epsilon)` for every `epsilon > 0`. So the Riemann hypothesis can be written as a promise about how fast one loop fills a surface.

## In the tree

[The Apollonian gasket note](/research/apollonian/#the-horocycle) proves that the Ford circles are exactly the images of the line `Im z = 1`, one closed horocycle on the modular surface, and reads off which Ford circles a lower line cuts. [The novelty meter](/research/stack/#the-novelty-meter) in the stack note reads the smoothed totient sum `sum_n phi(n) f(n t)`, built from the same `phi(c)` counts of fractions, and hears the zeros through the same exponent `3/4`, read in `q = t^2`; [the paper](/papers/novelty-meter/) writes it out and [the novelty demo](/demos/novelty/) plays it live.