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In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem.
We briefly present some facts from hyperbolic geometry which are helpful in understanding prime geodesics.
Consider the Poincaré half-plane model H of 2-dimensional hyperbolic geometry. Given a Fuchsian group, that is, a discrete subgroup Γ of PSL(2, R), Γ acts on H via linear fractional transformation. Each element of PSL(2, R) in fact defines an isometry of H, so Γ is a group of isometries of H.
There are then 3 types of transformation: hyperbolic, elliptic, and parabolic. (The loxodromic transformations are not present because we are working with real numbers.) Then an element γ of Γ has 2 distinct real fixed points if and only if γ is hyperbolic. See Classification of isometries and Fixed points of isometries for more details.
Now consider the quotient surface M=Γ\H. The following description refers to the upper half-plane model of the hyperbolic plane. This is a hyperbolic surface, in fact, a Riemann surface. Each hyperbolic element h of Γ determines a closed geodesic of Γ\H: first, by connecting the geodesic semicircle joining the fixed points of h, we get a geodesic on H called the axis of h, and by projecting this geodesic to M, we get a geodesic on Γ\H.
This geodesic is closed because 2 points which are in the same orbit under the action of Γ project to the same point on the quotient, by definition.
It can be shown that this gives a 1-1 correspondence between closed geodesics on Γ\H and hyperbolic conjugacy classes in Γ. The prime geodesics are then those geodesics that trace out their image exactly once — algebraically, they correspond to primitive hyperbolic conjugacy classes, that is, conjugacy classes {γ} such that γ cannot be written as a nontrivial power of another element of Γ.
The importance of prime geodesics comes from their relationship to other branches of mathematics, especially dynamical systems, ergodic theory, and number theory, as well as Riemann surfaces themselves. These applications often overlap among several different research fields.
In dynamical systems, the closed geodesics represent the periodic orbits of the geodesic flow.
In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the prime number theorem. To be specific, we let π(x) denote the number of closed geodesics whose norm (a function related to length) is less than or equal to x; then π(x) ~ x/ln(x). This result is usually credited to Atle Selberg. In his 1970 Ph.D. thesis, Grigory Margulis proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak proved an analogue of Chebotarev's density theorem.
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