Farey map, Diophantine approximation and Bruhat-Tits tree

2014/01/23 by Dong Han Kim, Seonhee Lim, Kim, Dong Han +5
Mathematics · Physics and Astronomy · #11J61 #11J70 #20E18 #37A45 #Advanced Mathematical Theories and Applications #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.GR #math.NT #msc:11J61 #msc:11J70 #msc:20E18 #msc:37A45

paper · pdf · doi:10.48550/arxiv.1401.5866

19 pages, 2 figures

arxiv created 2014/01/23 · openalex publication_date 2014/01/23 · arxiv updated 2014/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.

Related

• Distribution of the periodic points of the Farey map
• An effective estimate for the Lebesgue measure of preimages of iterates of the Farey map
• A slow triangle map with a segment of indifferent fixed points and a\n complete tree of rational pairs
• The Farey Sequence, Stern Brocot Tree and Euclids Theorem
• The Farey map exploited in wavelet analysis -- Construction of an associated Farey mother wavelet