# American Institute of Mathematical Sciences

October  2012, 6(4): 539-561. doi: 10.3934/jmd.2012.6.539

 1 Boston College, Department of Mathematics, Chestnut Hill, MA 02467, United States

Received  June 2012 Published  January 2013

A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of closed geodesics having a prescribed linking number become equidistributed on average with respect to the Liouville measure. We show this by using the thermodynamic formalism to prove an equidistribution result for a corresponding set of quadratic irrationals on the unit interval.
Citation: Dubi Kelmer. Quadratic irrationals and linking numbers of modular knots. Journal of Modern Dynamics, 2012, 6 (4) : 539-561. doi: 10.3934/jmd.2012.6.539
##### References:
 [1] E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Hamb. Math. Abh., 3 (1924), 170-177. [2] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math., 92 (1988), 73-90. doi: 10.1007/BF01393993. [3] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$, Invent. Math., 114 (1993), 207-218. doi: 10.1007/BF01232667. [4] É. Ghys, Knots and dynamics, in "International Congress of Mathematicians," Vol. I, Eur. Math. Soc., Zürich, (2007), 247-277. doi: 10.4171/022-1/11. [5] D. A. Hejhal, "The Selberg Trace Formula for $PSL(2, \mathbf R)$," Vol. 2, Lecture Notes in Mathematics, Vol. 1001, Springer-Verlag, Berlin, 1983. [6] T. Kato, "A Short Introduction to Perturbation Theory for Linear Operators," Springer-Verlag, New York-Berlin, 1982. [7] J. Korevaar, "Tauberian Theory. A Century of Developments," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 329, Springer-Verlag, Berlin, 2004. [8] A. Katsuda and T. Sunada, Closed orbits in homology classes, Inst. Hautes Études Sci. Publ. Math., No. 71 (1990), 5-32. [9] S. P. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J., 58 (1989), 795-821. doi: 10.1215/S0012-7094-89-05837-7. [10] D. H. Mayer, On a $\zeta$ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203. [11] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55-60. doi: 10.1090/S0273-0979-1991-16023-4. [12] C. J. Mozzochi, Linking numbers of modular geodesics, preprint, (2010). [13] M. Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France, 114 (1986), 431-446. [14] _____, Homology and closed geodesics in a compact negatively curved surface, Amer. J. Math., 113 (1991), 379-385. doi: 10.2307/2374830. [15] R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math. J., 55 (1987), 287-297. doi: 10.1215/S0012-7094-87-05515-3. [16] H. Rademacher and E. Grosswald, "Dedekind Sums," The Carus Mathematical Monographs, No. 16, The Mathematical Association of America, Washington, D. C., 1972. [17] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory, 15 (1982), 229-247. doi: 10.1016/0022-314X(82)90028-2. [18] _____, Reciprocal geodesics, in "Analytic Number Theory," Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, (2007), 217-237. [19] _____, Linking numbers of modular knots, Commun. Math. Anal., 8 (2010), 136-144. [20] C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69. [21] S. Zelditch, Trace formula for compact $\Gamma\backslash PSL_2(\mathbf R)$ and the equidistribution theory of closed geodesics, Duke Math. J., 59 (1989), 27-81. doi: 10.1215/S0012-7094-89-05902-4.

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##### References:
 [1] E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Hamb. Math. Abh., 3 (1924), 170-177. [2] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math., 92 (1988), 73-90. doi: 10.1007/BF01393993. [3] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$, Invent. Math., 114 (1993), 207-218. doi: 10.1007/BF01232667. [4] É. Ghys, Knots and dynamics, in "International Congress of Mathematicians," Vol. I, Eur. Math. Soc., Zürich, (2007), 247-277. doi: 10.4171/022-1/11. [5] D. A. Hejhal, "The Selberg Trace Formula for $PSL(2, \mathbf R)$," Vol. 2, Lecture Notes in Mathematics, Vol. 1001, Springer-Verlag, Berlin, 1983. [6] T. Kato, "A Short Introduction to Perturbation Theory for Linear Operators," Springer-Verlag, New York-Berlin, 1982. [7] J. Korevaar, "Tauberian Theory. A Century of Developments," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 329, Springer-Verlag, Berlin, 2004. [8] A. Katsuda and T. Sunada, Closed orbits in homology classes, Inst. Hautes Études Sci. Publ. Math., No. 71 (1990), 5-32. [9] S. P. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J., 58 (1989), 795-821. doi: 10.1215/S0012-7094-89-05837-7. [10] D. H. Mayer, On a $\zeta$ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203. [11] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55-60. doi: 10.1090/S0273-0979-1991-16023-4. [12] C. J. Mozzochi, Linking numbers of modular geodesics, preprint, (2010). [13] M. Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France, 114 (1986), 431-446. [14] _____, Homology and closed geodesics in a compact negatively curved surface, Amer. J. Math., 113 (1991), 379-385. doi: 10.2307/2374830. [15] R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math. J., 55 (1987), 287-297. doi: 10.1215/S0012-7094-87-05515-3. [16] H. Rademacher and E. Grosswald, "Dedekind Sums," The Carus Mathematical Monographs, No. 16, The Mathematical Association of America, Washington, D. C., 1972. [17] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory, 15 (1982), 229-247. doi: 10.1016/0022-314X(82)90028-2. [18] _____, Reciprocal geodesics, in "Analytic Number Theory," Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, (2007), 217-237. [19] _____, Linking numbers of modular knots, Commun. Math. Anal., 8 (2010), 136-144. [20] C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69. [21] S. Zelditch, Trace formula for compact $\Gamma\backslash PSL_2(\mathbf R)$ and the equidistribution theory of closed geodesics, Duke Math. J., 59 (1989), 27-81. doi: 10.1215/S0012-7094-89-05902-4.
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