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Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Applications to Non-Archimedean Diophantine Approximation

Birkhäuser

Authors:

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  • 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)

  1. Front Matter

    Pages i-viii
  2. Introduction

    • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
    Pages 1-19
  3. Geometry and Dynamics in Negative Curvature

    1. Front Matter

      Pages 21-21
    2. Negatively Curved Geometry

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 23-48
    3. Potentials, Critical Exponents,and Gibbs Cocycles

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 49-82
    4. Patterson–Sullivan and Bowen–Margulis Measures with Potential on CAT(–1) Spaces

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 83-110
    5. Symbolic Dynamics of Geodesic Flows on Trees

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 111-140
    6. Random Walks on Weighted Graphs of Groups

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 141-154
    7. Skinning Measures with Potential on CAT(–1) Spaces

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 155-167
    8. Explicit Measure Computations for Simplicial Trees and Graphs of Groups

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 169-180
    9. Rate of Mixing for the Geodesic Flow

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 181-203
  4. Geometric Equidistribution and Counting

    1. Front Matter

      Pages 205-205
    2. Equidistribution of Equidistant Level Sets to Gibbs Measures

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 207-224
    3. Equidistribution of Common Perpendicular Arcs

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 225-252
    4. Equidistribution and Counting of Common Perpendiculars in Quotient Spaces

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 253-285
    5. Geometric Applications

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 287-296
  5. Arithmetic Applications

    1. Front Matter

      Pages 297-297
    2. Fields with Discrete Valuations

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 299-305
    3. Bruhat–Tits Trees and Modular Groups

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 307-328
    4. Equidistribution and Counting of Rational Points in Completed Function Fields

      • Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
      Pages 329-340

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.

Keywords

  • equidistribution
  • orbit counting
  • geodesic flow
  • negative curvature
  • common perpendicular
  • ortholength spectrum
  • equilibrium state
  • Gibbs measure
  • skinning measure

Authors and Affiliations

  • Laboratoire de mathématique d’Orsay, Université Paris-Saclay, Orsay, France

    Anne Broise-Alamichel, Frédéric Paulin

  • Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Jouni Parkkonen

Bibliographic Information

Buying options

eBook
USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-18315-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 159.99
Price excludes VAT (USA)
Hardcover Book
USD 159.99
Price excludes VAT (USA)