Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees: Applications to Non-Archimedean Diophantine Approximation (Progress in Mathematics Book 329)

"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"--Publisher's description Contents......Page 6 Chapter 1 Introduction......Page 10 1.1 Geometric and dynamical tools......Page 11 1.2 The distribution of common perpendiculars......Page 15 1.3 Counting in weighted graphs of groups......Page 20 1.4 Selected arithmetic applications......Page 24 1.5 General notation......Page 28 Part I Geometry and Dynamics in Negative Curvature......Page 29 2.1 Background on CAT(-1) spaces......Page 30 2.2 Generalised geodesic lines......Page 36 2.3 The unit tangent bundle......Page 38 2.4 Normal bundles and dynamical neighbourhoods......Page 43 2.5 Creating common perpendiculars......Page 47 2.6 Metric and simplicial trees, and graphs of groups......Page 48 Discrete-time geodesic flow on trees......Page 50 Bass–Serre's graphs of groups......Page 51 3.1 Background on (uniformly local) Hölder continuity......Page 56 3.2 Potentials......Page 72 3.3 Poincaré series and critical exponents......Page 77 3.4 Gibbs cocycles......Page 81 3.5 Systems of conductances on trees and generalised electrical networks......Page 86 4.1 Patterson densities......Page 90 4.2 Gibbs measures......Page 94 The Gibbs property of Gibbs measures......Page 95 The Hopf–Tsuji–Sullivan–Roblin theorem......Page 96 On the finiteness of Gibbs measures......Page 97 Bowen–Margulis measure computations in locally symmetric spaces......Page 99 On the cohomological invariance of Gibbs measures......Page 102 4.3 Patterson densities for simplicial trees......Page 104 4.4 Gibbs measures for metric and simplicial trees......Page 107 5.1 Two-sided topological Markov shifts......Page 118 5.2 Coding discrete-time geodesic flows on simplicial trees......Page 119 5.3 Coding continuous-time geodesic flows on metric trees......Page 132 5.4 The variational principle for metric and simplicial trees......Page 139 6.1 Laplacian operators on weighted graphs of groups......Page 148 6.2 Patterson densities as harmonic measures for simplicial trees......Page 154 7.1 Skinning measures......Page 162 7.2 Equivariant families of convex subsets and their skinning measures......Page 172 Chapter 8 Explicit Measure Computations for Simplicial Trees and Graphs of Groups......Page 175 8.1 Computations of Bowen–Margulis measures for simplicial trees......Page 176 8.2 Computations of skinning measures for simplicial trees......Page 181 9.1 Rate of mixing for Riemannian manifolds......Page 187 9.2 Rate of mixing for simplicial trees......Page 188 9.3 Rate of mixing for metric trees......Page 200 Part II Geometric Equidistribution and Counting......Page 210 10.1 A general equidistribution result......Page 211 10.2 Rate of equidistribution of equidistant level sets for manifolds......Page 217 10.3 Equidistribution of equidistant level sets on simplicial graphs and random walks on graphs of groups......Page 219 10.4 Rate of equidistribution for metric and simplicial trees......Page 224 Chapter 11 Equidistribution of Common Perpendicular Arcs......Page 229 11.1 Part I of the proof of Theorem 11.1: The common part......Page 232 11.2 Part II of the proof of Theorem 11.1: The metric tree case......Page 234 11.3 Part III of the proof of Theorem 11.1: The manifold case......Page 238 11.4 Equidistribution of common perpendiculars in simplicial trees......Page 246 Chapter 12 Equidistribution and Counting of Common Perpendiculars in Quotient Spaces......Page 257 12.1 Multiplicities and counting functions in Riemannian orbifolds......Page 258 12.2 Common perpendiculars in Riemannian orbifolds......Page 260 12.3 Error terms for equidistribution and counting for Riemannian orbifolds......Page 264 12.4 Equidistribution and counting for quotient simplicial and metric trees......Page 268 12.5 Counting for simplicial graphs of groups......Page 275 12.6 Error terms for equidistribution and counting for metric and simplicial graphs of groups......Page 282 13.1 Orbit counting in conjugacy classes for groups acting on trees......Page 290 13.2 Equidistribution and counting of closed orbits on metric and simplicial graphs (of groups)......Page 294 Part III Arithmetic Applications......Page 300 14.1 Local fields and valuations......Page 301 14.2 Global function fields......Page 303 15.1 Bruhat–Tits trees......Page 308 15.2 Modular graphs of groups......Page 312 15.3 Computations of measures for Bruhat–Tits trees......Page 314 15.4 Exponential decay of correlation and error terms for arithmetic quotients of Bruhat–Tits trees......Page 319 15.5 Geometrically finite lattices with infinite Bowen–Margulis measure......Page 326 16.1 Counting and equidistribution of non-Archimedean Farey fractions......Page 330 16.2 Mertens's formula in function fields......Page 339 Chapter 17 Equidistribution and Counting of Quadratic Irrational Points in Non-Archimedean Local Fields......Page 342 17.1 Counting and equidistribution of loxodromic fixed points......Page 343 17.2 Counting and equidistribution of quadratic irrationals in positive characteristic......Page 347 17.3 Counting and equidistribution of quadratic irrationals in Qp......Page 355 Chapter 18 Equidistribution and Counting of Cross-ratios......Page 361 18.1 Counting and equidistribution of cross-ratios of loxodromic fixed points......Page 362 18.2 Counting and equidistribution of cross-ratios of quadratic irrationals......Page 368 Chapter 19 Equidistribution and Counting of Integral Representations by Quadratic Norm Forms......Page 371 A.1 Introduction......Page 376 A.2 Proof of the main result, Theorem A.4......Page 380 List of Symbols......Page 386 Bibliography......Page 393 Index......Page 407