scholarly journals Local rigidity of actions of higher rank abelian groups and KAM method

2004 ◽  
Vol 10 (16) ◽  
pp. 142-154 ◽  
Author(s):  
Danijela Damjanović ◽  
Anatole Katok
Keyword(s):  
Abelian Groups ◽  
Local Rigidity ◽  
Higher Rank ◽  
Kam Method

Local rigidity of Anosov higher-rank lattice actions

1998 ◽  
Vol 18 (3) ◽  
pp. 687-702 ◽  
Author(s):  
NANTIAN QIAN ◽  
CHENGBO YUE
Keyword(s):  
Abelian Group ◽  
Lyapunov Functional ◽  
Rigidity Result ◽  
Absolute Value ◽  
Local Rigidity ◽  
Higher Rank ◽  
Standard Action ◽  
Lattice Actions

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

scholarly journals The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups

2014 ◽  
Vol 24 (4) ◽  
pp. 1204-1228 ◽  
Author(s):  
Anatole Katok ◽  
Svetlana Katok ◽  
Federico Rodriguez Hertz
Keyword(s):  
Abelian Groups ◽  
Higher Rank

scholarly journals Local rigidity of affine actions of higher rank groups and lattices

2009 ◽  
Vol 170 (1) ◽  
pp. 67-122 ◽  
Author(s):  
David Fisher ◽  
Gregory Margulis
Keyword(s):  
Local Rigidity ◽  
Higher Rank ◽  
Affine Actions

scholarly journals Arithmeticity and topology of smooth actions of higher rank abelian groups

2016 ◽  
Vol 10 (02) ◽  
pp. 135-172 ◽  
Author(s):  
Anatole Katok ◽  
Federico Rodriguez Hertz
Keyword(s):  
Abelian Groups ◽  
And Topology ◽  
Higher Rank

Local rigidity of higher rank non-abelian action on torus

2017 ◽  
Vol 39 (06) ◽  
pp. 1668-1709
Author(s):  
ZHENQI JENNY WANG
Keyword(s):  
Iterative Scheme ◽  
Local Rigidity ◽  
Rank One ◽  
Higher Rank ◽  
Toral Automorphisms

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.

scholarly journals Topological realizations and fundamental groups of higher-rank graphs

2015 ◽  
Vol 59 (1) ◽  
pp. 143-168 ◽  
Author(s):  
S. Kaliszewski ◽  
Alex Kumjian ◽  
John Quigg ◽  
Aidan Sims
Keyword(s):  
Fundamental Group ◽  
Abelian Groups ◽  
Fundamental Groups ◽  
Crossed Products ◽  
Finitely Generated ◽  
Topological Realization ◽  
Category Equivalence ◽  
Free Abelian Groups ◽  
Higher Rank ◽  
Projective Limits

AbstractWe investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graphΛ, this functor determines a category equivalence between the category of coverings ofΛand the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions fork-graphs: projective limits and crossed products by finitely generated free abelian groups.

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