Equidistribution and counting under equilibrium states in negative curvature and trees : applications to non-Archimedean diophantine approximation / Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin ; appendix by Jérôme Buzzi.

Format:

Book

Author/Creator:

Broise-Alamichel, Anne, author.

Parkkonen, Jouni, author.

Paulin, Frédéric, author.

Contributor:

Buzzi, Jérôme, writer of supplementary textual content.

Series:

Progress in mathematics (Boston, Mass.) ; v. 329.

Progress in Mathematics, 0743-1643 ; Volume 329

Language:

English

Subjects (All):

Geodesics (Mathematics).

Ergodic theory.

Dynamics.

Geometric group theory.

Physical Description:

viii, 413 pages : illustrations ; 25 cm.

Place of Publication:

Cham, Switzerland : Birkhäuser/Springer Nature Switzerland AG, [2019]

Summary:

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.

Contents:

Introduction

Negatively curved geometry

Potentials, critical exponents and Gibbs cocycles

Patterson-Sullivan and Bowen-Margulis measures with potential on CAT(-1) spaces

Symbolic dynamics of geodesic flows on trees

Random walks on weighted graphs of groups

Skinning measures with potential on CAT(-1) spaces

Explicit measure computations for simplicial trees and graphs of groups

Rate of mixing for the geodesic flow

Equidistribution of equidistant level sets to Gibbs measures

Equidistribution of common perpendicular arcs

Equidistribution and counting of common perpendiculars in quotient spaces

Geometric applications

Fields with discrete valuations

Bruhat-Tits trees and modular groups

Rational point equidistribution and counting in completed function fields

Equidistribution and counting of quadratic irrational points in non-Archimedean local fields

Counting and equidistribution of crossratios

Counting and equidistribution of integral representations by quadratic norm forms

A

A weak Gibbs measure is the unique equilibrium, by J. Buzzi

List of Symbols

Index

Bibliography.

Notes:

Includes bibliographical references (pages 395-408) and index.

Current copyright fee: GBP34.00 22\1.

Other Format:

Online version: Broise-Alamichel, Anne. Equidistribution and counting under equilibrium states in negative curvature and trees.

ISBN:

9783030183141

3030183149

OCLC:

1090485115