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

Quadratic irrationals and linking numbers of modular knots

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.   Google Scholar

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W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms,, Invent. Math., 92 (1988), 73.  doi: 10.1007/BF01393993.  Google Scholar

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I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$,, Invent. Math., 114 (1993), 207.  doi: 10.1007/BF01232667.  Google Scholar

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É. Ghys, Knots and dynamics,, in, (2007), 247.  doi: 10.4171/022-1/11.  Google Scholar

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D. A. Hejhal, "The Selberg Trace Formula for $PSL(2, \mathbf R)$," Vol. 2,, Lecture Notes in Mathematics, (1001).   Google Scholar

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T. Kato, "A Short Introduction to Perturbation Theory for Linear Operators,", Springer-Verlag, (1982).   Google Scholar

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J. Korevaar, "Tauberian Theory. A Century of Developments,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2004).   Google Scholar

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A. Katsuda and T. Sunada, Closed orbits in homology classes,, Inst. Hautes Études Sci. Publ. Math., (1990), 5.   Google Scholar

[9]

S. P. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature,, Duke Math. J., 58 (1989), 795.  doi: 10.1215/S0012-7094-89-05837-7.  Google Scholar

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D. H. Mayer, On a $\zeta$ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195.   Google Scholar

[11]

_____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55.  doi: 10.1090/S0273-0979-1991-16023-4.  Google Scholar

[12]

C. J. Mozzochi, Linking numbers of modular geodesics,, preprint, (2010).   Google Scholar

[13]

M. Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals,, Bull. Soc. Math. France, 114 (1986), 431.   Google Scholar

[14]

_____, Homology and closed geodesics in a compact negatively curved surface,, Amer. J. Math., 113 (1991), 379.  doi: 10.2307/2374830.  Google Scholar

[15]

R. Phillips and P. Sarnak, Geodesics in homology classes,, Duke Math. J., 55 (1987), 287.  doi: 10.1215/S0012-7094-87-05515-3.  Google Scholar

[16]

H. Rademacher and E. Grosswald, "Dedekind Sums,", The Carus Mathematical Monographs, (1972).   Google Scholar

[17]

P. Sarnak, Class numbers of indefinite binary quadratic forms,, J. Number Theory, 15 (1982), 229.  doi: 10.1016/0022-314X(82)90028-2.  Google Scholar

[18]

_____, Reciprocal geodesics,, in, 7 (2007), 217.   Google Scholar

[19]

_____, Linking numbers of modular knots,, Commun. Math. Anal., 8 (2010), 136.   Google Scholar

[20]

C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69.  doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[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.  doi: 10.1215/S0012-7094-89-05902-4.  Google Scholar

show all references

References:
[1]

E. Artin, Ein mechanisches System mit quasiergodischen Bahnen,, Hamb. Math. Abh., 3 (1924), 170.   Google Scholar

[2]

W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms,, Invent. Math., 92 (1988), 73.  doi: 10.1007/BF01393993.  Google Scholar

[3]

I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$,, Invent. Math., 114 (1993), 207.  doi: 10.1007/BF01232667.  Google Scholar

[4]

É. Ghys, Knots and dynamics,, in, (2007), 247.  doi: 10.4171/022-1/11.  Google Scholar

[5]

D. A. Hejhal, "The Selberg Trace Formula for $PSL(2, \mathbf R)$," Vol. 2,, Lecture Notes in Mathematics, (1001).   Google Scholar

[6]

T. Kato, "A Short Introduction to Perturbation Theory for Linear Operators,", Springer-Verlag, (1982).   Google Scholar

[7]

J. Korevaar, "Tauberian Theory. A Century of Developments,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2004).   Google Scholar

[8]

A. Katsuda and T. Sunada, Closed orbits in homology classes,, Inst. Hautes Études Sci. Publ. Math., (1990), 5.   Google Scholar

[9]

S. P. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature,, Duke Math. J., 58 (1989), 795.  doi: 10.1215/S0012-7094-89-05837-7.  Google Scholar

[10]

D. H. Mayer, On a $\zeta$ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195.   Google Scholar

[11]

_____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55.  doi: 10.1090/S0273-0979-1991-16023-4.  Google Scholar

[12]

C. J. Mozzochi, Linking numbers of modular geodesics,, preprint, (2010).   Google Scholar

[13]

M. Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals,, Bull. Soc. Math. France, 114 (1986), 431.   Google Scholar

[14]

_____, Homology and closed geodesics in a compact negatively curved surface,, Amer. J. Math., 113 (1991), 379.  doi: 10.2307/2374830.  Google Scholar

[15]

R. Phillips and P. Sarnak, Geodesics in homology classes,, Duke Math. J., 55 (1987), 287.  doi: 10.1215/S0012-7094-87-05515-3.  Google Scholar

[16]

H. Rademacher and E. Grosswald, "Dedekind Sums,", The Carus Mathematical Monographs, (1972).   Google Scholar

[17]

P. Sarnak, Class numbers of indefinite binary quadratic forms,, J. Number Theory, 15 (1982), 229.  doi: 10.1016/0022-314X(82)90028-2.  Google Scholar

[18]

_____, Reciprocal geodesics,, in, 7 (2007), 217.   Google Scholar

[19]

_____, Linking numbers of modular knots,, Commun. Math. Anal., 8 (2010), 136.   Google Scholar

[20]

C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69.  doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[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.  doi: 10.1215/S0012-7094-89-05902-4.  Google Scholar

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