scholarly journals Global rigidity of higher rank Anosov actions on tori and nilmanifolds

2012 ◽  
Vol 26 (1) ◽  
pp. 167-198 ◽  
Author(s):  
David Fisher ◽  
Boris Kalinin ◽  
Ralf Spatzier
Keyword(s):  
Global Rigidity ◽  
Anosov Actions ◽  
Higher Rank

scholarly journals Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

2018 ◽  
Vol 40 (1) ◽  
pp. 117-141 ◽  
Author(s):  
DANIJELA DAMJANOVIĆ ◽  
DISHENG XU
Keyword(s):  
Fixed Point ◽  
Universal Cover ◽  
Diffeomorphism Group ◽  
Finite Cover ◽  
Diffeomorphism Groups ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Rigidity Results ◽  
Anosov Actions ◽  
Higher Rank

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices

2001 ◽  
Vol 21 (1) ◽  
pp. 121-164 ◽  
Author(s):  
GREGORY A. MARGULIS ◽  
NANTIAN QIAN
Keyword(s):  
Lie Groups ◽  
Left Translation ◽  
Real Rank ◽  
Semisimple Lie Groups ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Local Rigidity ◽  
Algebraic Actions ◽  
Anosov Actions ◽  
Higher Rank

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

Non-abelian cohomology of abelian Anosov actions

2000 ◽  
Vol 20 (1) ◽  
pp. 259-288 ◽  
Author(s):  
ANATOLE KATOK ◽  
VIOREL NIŢICĂ ◽  
ANDREI TÖRÖK
Keyword(s):  
Discrete Time ◽  
Lie Group ◽  
Global Information ◽  
Infinite Dimensional ◽  
Anosov Actions ◽  
Geometric Character ◽  
Higher Rank ◽  
Real Analytic ◽  
A New Technique ◽  
Discrete Time Case

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

scholarly journals Global rigidity of higher rank abelian Anosov algebraic actions

2014 ◽  
Vol 198 (1) ◽  
pp. 165-209 ◽  
Author(s):  
Federico Rodriguez Hertz ◽  
Zhiren Wang
Keyword(s):  
Global Rigidity ◽  
Algebraic Actions ◽  
Higher Rank

scholarly journals Rigidity for Anosov Actions of Higher Rank Lattices

1992 ◽  
Vol 135 (2) ◽  
pp. 361 ◽  
Author(s):  
Steven Hurder
Keyword(s):  
Anosov Actions ◽  
Higher Rank

scholarly journals Invariant measures for higher-rank hyperbolic abelian actions

1996 ◽  
Vol 16 (4) ◽  
pp. 751-778 ◽  
Author(s):  
A. Katok ◽  
R. J. Spatzier
Keyword(s):  
Invariant Measures ◽  
Metric Entropy ◽  
Ergodic Measures ◽  
Anosov Actions ◽  
Higher Rank ◽  
Partially Hyperbolic

AbstractWe investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤkandWe show that they are either Haar measures or that every element of the action has zero metric entropy.

scholarly journals On classification of higher rank Anosov actions on compact manifold

2020 ◽  
Vol 238 (2) ◽  
pp. 745-806
Author(s):  
Danijela Damjanović ◽  
Disheng Xu
Keyword(s):  
Compact Manifold ◽  
Anosov Actions ◽  
Higher Rank
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