Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees: Applications to Non-Archimedean Diophantine Approximation

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