Prove: Let S is a closed hyperbolic surface. Then the geodesic flow on S has only finitely many closed orbits of period less than any given number.

Question

I know it's equivalent to the discreteness of length spectrum,but how to prove it using geodesic flow ?

I'm Chinese and sorry for my poor English.

Many thanks.

Answer 1

The simplest way is to note that after considering a Markov system for a hyperbolic flow (although there are specific codings in the particular case of the geodesic flow on constant negative curvature, and any closed hyperbolic surface obtained from making the quotient), the closed geodesics (except possibly finitely many of them) are in a one-to-one correspondence to the periodic orbits on the base of the suspension.

Since the periodic orbits on the base of a given period are finite, you get your statement because the rectangles of the Markov system are bounded (and thus the time that the closed orbits may travel between boundaries is also bounded).

For full details see for example in https://galton.uchicago.edu/~lalley/Papers/homology.pdf. This will be complicated or not depending on what is your background. It should be sufficient to look only at page 798.

Answer 2

There is a very soft proof which does not use much:

First of all, by the very definition, closed trajectories of geodesic flow are in bijective correspondence with closed geodesics in the surface itself and the length stays the same (just project your trajectory down to the surface). Now, suppose you have an infinite sequence of closed geodesics $c_n$ (with the unit speed) in a compact hyperbolic surface $S$ of length $L_n\le L_0$. Think of each $c_n$ as a map $c_n: [0,L_n]\to S$. Now, use the Arzela- Ascoli theorem, since the family is clearly equicontinuous and the target is compact. It follows that after passing to a subsequence, there is a limit $\lim_n c_n= c, c:L\to S$. Next, check that $c$ is a local distance-minimizer which implies that $c$ is again a geodesic. By continuity, $c$ is periodic: $c(0)=c(L), c'(0)=c'(L)$. Next, use the straight-line homotopy to show that $c_n$ is freely homotopic to $c$ for all large $n$. But in a hyperbolic surface freely homotopic closed geodesics have to have equal image and differ only by a reparameterization. (The easiest way to see that is to consider the elements of the fundamental group corresponding to these geodesics, identified with isometries of the hyperbolic plane: These isometries have to be equal to a translation along two hyperbolic geodesics. But a hyperbolic translation can have only one invariant line.) Thus, $c_n=c$ for all large $n$. Hence, our sequence of geodesics is finite. qed