Abstract
A closed geodesic on the modular surface gives rise to a knot on the3-sphere with a trefoil knot removed, and one can compute the linkingnumber of such a knot with the trefoil knot. We show that, whenordered by their length, the set of closed geodesics having aprescribed linking number become equidistributed on average withrespect to the Liouville measure. We show this by using thethermodynamic formalism to prove an equidistribution result for acorresponding set of quadratic irrationals on the unit interval.
Highlights
Let M = PSL2(Z)\H denote the modular surface; here H denotes the upper half plane endowed with the hyperbolic metric, and PSL2(Z) acts by isometries on H via linear fractional transformation
Following the results of Ghys [Gh07] and Sarnak [Sa10] on the linking numbers of modular knots, we continue the study of the set of primitive closed geodesics having a prescribed linking number, and in particular their distribution on T 1M
In [Gh07], Ghys showed that T 1M is homeomorphic to the threesphere without a trefoil knot, and a closed geodesic can be thought of as a knot in this space
Summary
Instead of ordering the geodesics by length one can order them according to their discriminant (that is, the discriminant of the corresponding quadratic form) In this setting, Duke’s theorem [Du88] shows that the set of closed geodesics of a given discriminant become equidistributed in T 1M as the discriminant goes to infinity. One should compare Theorem 3 to analogous results on the counting and equidistribution of closed geodesics on a compact hyperbolic surface lying in a prescribed homology class In this case, Katsuda and Sunada [KS90], Lalley [La89], and Pollicott [Po91] obtained similar results using an entirely dynamical approach (which works in variable negative curvature). We note that the proof we give here is entirely dynamical and does not rely on the Selberg trace formula nor on Duke’s theorem
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