On Cross Sections to the Geodesic and Horocycle Flows on Quotients of\n \SL(2, \ℝ) by Hecke Triangle Groups G_q

2019/06/17 by Diaaeldin Taha, Taha, Diaaeldin
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1906.07250

openalex publication_date 2019/06/17 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

In this paper, we provide a model for cross sections to the geodesic and\nhorocycle flows on \SL(2, \ℝ)/G_q using an extension of\na heuristic of P. Arnoux and A. Nogueira. Our starting point is a continued\nfraction algorithm related to the group G_q, and a cross section to the\nhorocycle flow on \SL(2, \ℝ)/G_q from a previous paper.\nAs an application, we get the natural extension and invariant measure for a\nsymmetric G_q-Farey interval map resulting from projectivizing the\naforementioned continued fraction algorithm.\n

Citations

• Natural extensions and Gauss measures for piecewise homographic continued fractions

Related

• Symbolic dynamics for the geodesic flow on Hecke surfaces
• Poincaré sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics
• A Poincaré section for horocycle flow on the space of lattices
• Nearest lambda_q-multiple fractions
• First return systems for some continued fraction maps