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Journal of Modern Dynamics (JMD)
 

The variance of arithmetic measures associated to closed geodesics on the modular surface

Pages: 271 - 309, Issue 2, April 2009      doi:10.3934/jmd.2009.3.271

 
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Wenzhi Luo - Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, United States (email)
Zeév Rudnick - Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States (email)
Peter Sarnak - Department of Mathematics, Princeton University, Fine Hall,Washington Road, Princeton, NJ 08544 and School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States (email)

Abstract: We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.

Keywords:  Arithmetic variance,modular surface, closed geodesics.
Mathematics Subject Classification:  Primary: 11F72; Secondary: 37D40.

Received: October 2008;      Available Online: May 2009.