1 Introduction ................................................. 1
2 Background on negatively curved manifolds ................... 13
2.1 Uniform local Holder-continuity ........................ 14
2.2 Boundary at infinity, isometries and the Busemann
cocycle ................................................ 15
2.3 The geometry of the unit tangent bundle ................ 18
2.4 Geodesic flow, (un)stable foliations and the
Hamenstadt distances ................................... 21
2.5 Some exercises in hyperbolic geometry .................. 25
2.6 Pushing measures by branched covers .................... 28
3 A Patterson-Sullivan theory for Gibbs states ................ 31
3.1 Potential functions and their periods .................. 31
3.2 The Poincare series and the critical exponent of
(Г, F) ................................................. 34
3.3 The Gibbs cocycle of (Г, F) ............................ 39
3.4 The potential gap of (Г, F) ............................ 44
3.5 The crossratio of (Г, F) ............................... 48
3.6 The Patterson densities of (Г, F) ...................... 51
3.7 The Gibbs states of (Г, F) ............................. 57
3.8 The Gibbs property of Gibbs states ..................... 59
3.9 Conditional measures of Gibbs states on strong
(un)stable leaves ...................................... 66
4 Critical exponent and Gurevich pressure ..................... 75
4.1 Counting orbit points and periodic geodesies ........... 75
4.2 Logarithmic growth of the orbital counting functions ... 77
4.3 Equality between critical exponent and Gurevich
pressure ............................................... 85
4.4 Critical exponent of Schottky semigroups ............... 87
5 A Hopf-Tsuji-Sullivan-Roblin theorem for Gibbs states ...... 105
5.1 Some geometric notation ............................... 105
5.2 The Hopf-Tsuji-Sullivan-Roblin theorem for Gibbs
states ................................................ 108
5.3 Uniqueness of Patterson densities and of Gibbs
states ................................................ 118
6 Thermodynamic formalism and equilibrium states ............. 123
6.1 Measurable partitions and entropy ..................... 125
6.2 Proof of the variational principle .................... 127
7 The Liouville measure as a Gibbs measure ................... 143
7.1 The Holder-continuity of the (un)stable Jacobian ...... 146
7.2 Absolute continuity of the strong unstable foliation .. 150
7.3 The Liouville measure as an equilibrium state ......... 152
7.4 The Liouville measure satisfies the Gibbs property .... 155
7.5 Conservative Liouville measures are Gibbs measures .... 159
8 Finiteness and mixing of Gibbs states ...................... 165
8.1 Babillot's mixing criterion for Gibbs states .......... 165
8.2 A finiteness criterion for Gibbs states ............... 168
9 Growth and equidistributionof orbits and periods ........... 175
9.1 Convergence of measures on the square product ......... 175
9.2 Counting orbit points of discrete groups .............. 183
9.3 Equidistribution and counting of periodic orbits of
the geodesic flow ..................................... 185
9.4 The case of infinite Gibbs measure .................... 196
10 The ergodic theory of the strong unstable foliation ........ 199
10.1 Quasi-invariant transverse measures ................... 199
10.2 Quasi-invariant measures on the space of horospheres .. 203
10.3 Classification of quasi-invariant transverse
measures for Wsu ...................................... 206
11 Gibbs states on Galois covers .............................. 233
11.1 Improving Mohsen's shadow lemma ....................... 234
11.2 The Fatou-Roblin radial convergence theorem ........... 238
11.3 Characters and critical exponents ..................... 240
11.4 Characters and Patterson densities .................... 246
11.5 Characters and the Hopf-Tsuji-Sullivan-Roblin
theorem ............................................... 249
11.6 Galois covers and critical exponents .................. 253
11.7 Classification of ergodic Patterson densities on
nilpotent covers ...................................... 257
List of symbols ............................................... 261
Index ......................................................... 265
Bibliography .................................................. 271
|
