scholarly journals Cohomological equation and cocycle rigidity of discrete parabolic actions

2019 ◽  
Vol 39 (7) ◽  
pp. 3969-4000
Author(s):  
James Tanis ◽  
◽  
Zhenqi Jenny Wang ◽  
Keyword(s):  
Cohomological Equation ◽  
Cocycle Rigidity

Smooth cocycle rigidity for expanding maps, and an application to Mostow rigidity

2002 ◽  
Vol 132 (3) ◽  
pp. 439-452 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Lie Group ◽  
Piecewise Smooth ◽  
Rigidity Theorem ◽  
Compact Lie Group ◽  
Expanding Maps ◽  
Cocycle Rigidity ◽  
Mostow Rigidity

We give a variation on the proof of Mostow's rigidity theorem, for certain hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie group-valued cocycles over piecewise smooth expanding Markov maps.

scholarly journals Cocycle rigidity of partially hyperbolic abelian actions with almost rank-one factors

2017 ◽  
Vol 39 (7) ◽  
pp. 2006-2016
Author(s):  
KURT VINHAGE
Keyword(s):  
Recent Progress ◽  
Universal Cover ◽  
Cycle Structure ◽  
Rank One ◽  
Partially Hyperbolic ◽  
Cocycle Rigidity

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank-one factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.

scholarly journals The cohomological equation and invariant distributions for horocycle maps

2012 ◽  
Vol 34 (1) ◽  
pp. 299-340 ◽  
Author(s):  
JAMES TANIS
Keyword(s):  
Compact Manifolds ◽  
Invariant Distributions ◽  
Cohomological Equation ◽  
Sobolev Estimates

AbstractWe study the invariant distributions for horocycle maps on $\Gamma \backslash SL(2, \mathbb {R})$and prove Sobolev estimates for the cohomological equation of horocycle maps. As an application, we obtain a rate of equidistribution for horocycle maps on compact manifolds.

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