Rigidity in higher rank Abelian group actions. Vol.1, Introduction and cocycle problem

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
1. Rigidity in dynamics......Page 9
3. Rigidity for actions of higher rank abelian groups......Page 10
4. Mechanisms of rigidity......Page 11
a. Algebraic actions......Page 12
b. Non-uniform measure rigidity......Page 13
7. Background, references, and other sources......Page 14
Acknowledgements......Page 15
Part I: Preliminaries from dynamics and analysis......Page 17
1.1 Group actions, conjugacy, and related notions......Page 19
1.2 Functorial constructions......Page 20
1.3 Principal bundles......Page 21
1.4 Cocycles......Page 24
1.5 Roots and Weyl chambers for linear actions......Page 27
1.6.1 The linear part......Page 28
1.6.2 Affine actions of …......Page 29
1.6.3 Homogeneous and double coset actions......Page 30
1.6.4 Invariant distributions and their integrability......Page 31
1.6.5 Resonances......Page 32
1.7 Measurable and non-uniform differentiable setting......Page 33
1.7.1 Multiplicative ergodic theorem......Page 34
1.7.2 Lyapunov hyperplanes and Weyl chambers......Page 35
1.7.3 Invariant manifolds......Page 37
1.8.1 Anosov diffeomorphisms and flows......Page 38
1.8.2 Partially hyperbolic diffeomorphisms......Page 42
1.8.3 Anosov actions of higher rank abelian groups......Page 44
2.1.1 Nilpotency of the ambient group......Page 47
2.1.2 Ergodic automorphisms of the torus......Page 48
Preliminaries on nilpotent Lie groups and lattices......Page 50
Hyperbolic automorphisms......Page 52
First examples......Page 53
More examples......Page 55
2.1.4 Anosov diffeomorphisms on infratori and infranilmanifolds......Page 56
2.2.1 On algebraic conjugacy......Page 61
2.2.2 The genuine higher rank condition......Page 63
2.2.3 Rigidity of genuinely higher rank actions......Page 64
2.2.4 Irreducible actions and units in number fields......Page 65
2.2.5 Cartan actions......Page 67
2.2.6 Symplectic actions on T4......Page 68
2.2.7 Genuinely partially hyperbolic actions......Page 70
2.2.8 Affine actions of the torus without fixed points......Page 76
2.2.9 Higher rank abelian actions on nilmanifolds......Page 80
2.3.2 Suspensions of automorphisms of tori and nilmanifolds......Page 85
Summary of Lie group theory......Page 86
Definition of Weyl chamber flow and hyperbolicity......Page 87
Weyl chamber flow on SL(n,R)......Page 90
Weyl chamber flow on Sp(n,R)......Page 91
Weyl chamber flow on SO(n, n,R)......Page 93
Weyl chamber flow on SU(m, n)......Page 94
2.3.5 Twisted Weyl chamber flows and further extensions......Page 97
2.3.6 Reduction of scalars......Page 98
2.4.1 Non-invertible examples on tori and nilmanifolds and relations to solenoids......Page 100
2.4.2 Automorphisms of other compact abelian groups......Page 102
3.1 Introduction......Page 108
3.2 Preparatory norm estimates......Page 109
3.3 Journe's theorem......Page 117
3.4 The Jacobian along the stable leaves of a partially hyperbolic diffeomorphism......Page 126
3.5 Smooth regularity by Fourier method......Page 130
3.6 Real analytic regularity by Fourier method......Page 134
3.7.2 A smooth regularity result......Page 136
Part II: Cocycles, cohomology, and rigidity......Page 139
4.1.1 The relevance of cocycles for dynamical systems questions......Page 141
4.1.2 Principal methods for classification of cocycles......Page 144
4.2.1 Extension along orbits: the Livshitz theorem......Page 146
4.2.2 Regularity of the transfer function: the first appearance of the extension along stable manifolds method......Page 149
4.2.3 Invariant foliations for vector-valued extensions......Page 153
4.2.4 The first appearance of the harmonic analysis argument......Page 159
4.3.1 An overview......Page 160
4.3.2 Accessibility of foliations......Page 161
4.3.3 Periodic cycle functionals......Page 166
4.4 Higher rank results for vector-valued cocycles......Page 171
4.4.1 Higher rank trick and vanishing of the first cohomology......Page 172
4.4.2 Harmonic analysis method......Page 173
4.4.3 Geometric approach and TNS actions......Page 180
4.4.4 Geometric approach for Weyl chamber flows......Page 186
4.4.5 Rigidity of abelian cocycles over Cartan actions and algebraic K-theory......Page 193
4.5 Cocycles over generic Anosov actions......Page 198
4.6 Twisted cocycles......Page 206
5.1 Cocycles with values in compact abelian groups......Page 210
5.2.1 An overview......Page 215
5.2.2 The center-bunching condition......Page 216
5.3.1 Livshitz’ theorem for cocycles with values in Lie groups......Page 218
5.3.2 Livshitz theorem for cocycles with values in certain diffeomorphism groups......Page 220
5.4.1 Cocycles with values in Lie groups: the C1 case......Page 227
5.4.3 Cocycles with values in diffeomorphism groups......Page 235
5.4.4 Invariant foliations for bundle extensions......Page 244
5.5.1 Cocycles with values in Lie groups......Page 248
5.5.2 Cocycles with values in diffeomorphism groups......Page 251
5.5.3 Lack of regularity in the absence of bunching......Page 252
5.6 Parry’s general cohomological result for cocycles with compact non-abelian range......Page 256
5.7 Lift of regularity for the transfer map from measurable to Holder......Page 259
5.8 Periodic cycle functionals......Page 262
5.9 Non-abelian cocycles over TNS actions......Page 267
5.10 Rigidity of non-abelian cocycles over Cartan actions: K-theory approach......Page 277
6.1 Introduction to higher cohomology of group actions......Page 284
6.2.2 Periodic orbits and orbits for the dual action by toral endomorphisms......Page 287
6.2.3 The L1 n-cohomology of Zk......Page 293
6.2.4 Growth estimates for the orbits of the dual action......Page 297
6.2.5 Proofs of Theorems 6.2.1 and 6.2.2......Page 303
6.3 Cohomology for Weyl chamber flows......Page 307
References......Page 310
Index......Page 319