Authors:
- Introduces innovative ergodic techniques to Diophantine approximation in non-Archimedean local fields
- Gives numerous first published error terms in geometric counting and equidistribution problems
- Bridges the gap between the equidistribution and counting results with potentials on negatively curved manifolds and the ones without potential on trees
Part of the book series: Progress in Mathematics (PM, volume 329)
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Table of contents (19 chapters)
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Front Matter
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Geometry and Dynamics in Negative Curvature
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Front Matter
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Geometric Equidistribution and Counting
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Front Matter
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Arithmetic Applications
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Front Matter
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About this book
This book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees—again without the need for any compactness or torsionfree assumptions.
In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms.
One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields.
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Authors and Affiliations
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Laboratoire de mathématique d’Orsay, Université Paris-Saclay, Orsay, France
Anne Broise-Alamichel, Frédéric Paulin
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Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
Jouni Parkkonen
Bibliographic Information
Book Title: Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees
Book Subtitle: Applications to Non-Archimedean Diophantine Approximation
Authors: Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
Series Title: Progress in Mathematics
DOI: https://doi.org/10.1007/978-3-030-18315-8
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-18314-1Published: 02 January 2020
Softcover ISBN: 978-3-030-18317-2Published: 26 August 2021
eBook ISBN: 978-3-030-18315-8Published: 16 December 2019
Series ISSN: 0743-1643
Series E-ISSN: 2296-505X
Edition Number: 1
Number of Pages: VIII, 413
Number of Illustrations: 44 b/w illustrations, 14 illustrations in colour
Topics: Dynamical Systems and Ergodic Theory, Differential Geometry, Group Theory and Generalizations, Number Theory, Convex and Discrete Geometry, Probability Theory and Stochastic Processes