Journal of Modern Dynamics (JMD)

Quadratic irrationals and linking numbers of modular knots

Pages: 539 - 561, Issue 4, October 2012      doi:10.3934/jmd.2012.6.539

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Dubi Kelmer - Boston College, Department of Mathematics, Chestnut Hill, MA 02467, United States (email)

Abstract: 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.

Keywords:  Closed geodesics, Modular surface, linking numbers, equidistribution.
Mathematics Subject Classification:  Primary: 58F17; Secondary: 11F72, 30B70, 58F20.

Received: June 2012;      Available Online: January 2013.