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Equidistribution and counting under ...

Broise-Alamichel, Anne.

Equidistribution and counting under equilibrium states in negative curvature and treesapplications to non-Archimedean diophantine approximation /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Equidistribution and counting under equilibrium states in negative curvature and treesby Anne Broise-Alamichel, Jouni Parkkonen, Frederic Paulin.
其他題名:	applications to non-Archimedean diophantine approximation /
作者:	Broise-Alamichel, Anne.
其他作者:	Parkkonen, Jouni.
出版者:	Cham :Springer International Publishing :2019.
面頁冊數:	viii, 413 p. :ill., digital ;24 cm.
Contained By:	Springer eBooks
標題:	Diophantine approximation.
電子資源:	https://doi.org/10.1007/978-3-030-18315-8
ISBN:	9783030183158$q(electronic bk.)

Equidistribution and counting under equilibrium states in negative curvature and treesapplications to non-Archimedean diophantine approximation /
Broise-Alamichel, Anne.

Equidistribution and counting under equilibrium states in negative curvature and treesapplications to non-Archimedean diophantine approximation /[electronic resource] :by Anne Broise-Alamichel, Jouni Parkkonen, Frederic Paulin. - Cham :Springer International Publishing :2019. - viii, 413 p. :ill., digital ;24 cm. - Progress in mathematics,v.3290743-1643 ;. - Progress in mathematics ;v.295..

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.

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.

ISBN: 9783030183158$q(electronic bk.)

Standard No.: 10.1007/978-3-030-18315-8doiSubjects--Topical Terms:

191006
Diophantine approximation.

LC Class. No.: QA242 / .B76 2019

Dewey Class. No.: 512.73

Equidistribution and counting under equilibrium states in negative curvature and treesapplications to non-Archimedean diophantine approximation /
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505 0 $a 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.
520 $a 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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