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Distribution of the Periodic Points of the Farey Map

  • Byron HeersinkEmail author
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Abstract

We expand the cross section of the geodesic flow in the tangent bundle of the modular surface given by Series to produce another section whose return map under the geodesic flow is a double cover of the natural extension of the Farey map. We use this cross section to extend the correspondence between the closed geodesics on the modular surface and the periodic points of the Gauss map to include the periodic points of the Farey map. Then, analogous to the work of Pollicott, we prove an equidistribution result for the periodic points of the Farey map when they are ordered according to the length of their corresponding closed geodesics.

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Notes

Acknowledgements

I thank my advisor Florin Boca as well as Claire Merriman for many helpful discussions which inspired this paper. I particularly thank Merriman for making me aware of the characterization of the periodic points of F. I also thank the referee for helpful suggestions that improved the presentation of this paper. I also acknowledge support from Department of Education Grant P200A090062, “University of Illinois GAANN Mathematics Fellowship Project.”

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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