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April 2021 Extremal parameters and their duals for boundary maps associated to Fuchsian groups
Adam Abrams
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Illinois J. Math. 65(1): 153-179 (April 2021). DOI: 10.1215/00192082-8827631

Abstract

We describe arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature using generalized Bowen–Series boundary maps, and their natural extensions, associated to co-compact torsion-free Fuchsian groups. If the boundary map parameters are extremal—that is, each is an endpoint of a geodesic that extends a side of the fundamental polygon—then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parametrized by this set. This construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. Moreover, each extremal parameter choice has a corresponding dual parameter choice such that the “past” of the arithmetic code of a geodesic is the “future” for the code using the dual parameter. This duality was observed for two classical parameter choices by Adler and Flatto; here we show constructively that every extremal parameter choice has a dual.

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Adam Abrams. "Extremal parameters and their duals for boundary maps associated to Fuchsian groups." Illinois J. Math. 65 (1) 153 - 179, April 2021. https://doi.org/10.1215/00192082-8827631

Information

Received: 31 January 2020; Revised: 12 August 2020; Published: April 2021
First available in Project Euclid: 16 December 2020

Digital Object Identifier: 10.1215/00192082-8827631

Subjects:
Primary: 20H10
Secondary: 37B10 , 37D40

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign

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Vol.65 • No. 1 • April 2021
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