scholarly journals Local rigidity of partially hyperbolic actions I. KAM method and ℤkactions on the torus

2010 ◽  
Vol 172 (3) ◽  
pp. 1805-1858 ◽  
Author(s):  
Danijela Damjanović ◽  
Anatole Katok
Keyword(s):  
Local Rigidity ◽  
Partially Hyperbolic ◽  
Kam Method

On local rigidity of partially hyperbolic affine ℤk actions

2019 ◽  
Vol 2019 (751) ◽  
pp. 1-26
Author(s):  
Danijela Damjanović ◽  
Bassam Fayad
Keyword(s):  
Linear Part ◽  
Full Measure ◽  
Local Rigidity ◽  
Rank One ◽  
Partially Hyperbolic ◽  
Affine Action

AbstractThe following dichotomy for affine \mathbb{Z}^{k} actions on the torus {\mathbb{T}}^{d}, k,d\in{\mathbb{N}}, is shown to hold: (i) The linear part of the action has no rank-one factors, and then the affine action is locally rigid. (ii) The linear part of the action has a rank-one factor, and then the affine action is locally rigid in a probabilistic sense if and only if the rank-one factors are trivial. Local rigidity in a probabilistic sense means that rigidity holds for a set of full measure of translation vectors in the rank-one factors.

scholarly journals Local rigidity of partially hyperbolic actions

2010 ◽  
Vol 4 (2) ◽  
pp. 271-327 ◽  
Author(s):  
Zhenqi Jenny Wang ◽  
Keyword(s):  
Local Rigidity ◽  
Partially Hyperbolic

The Topology of Symplectic Partially Hyperbolic Automorphisms of the 4-Torus

2020 ◽  
Vol 108 (3-4) ◽  
pp. 462-464
Author(s):  
L. M. Lerman ◽  
K. N. Trifonov
Keyword(s):  
Partially Hyperbolic

scholarly journals Diffeomorphism cocycles over partially hyperbolic systems

2020 ◽  
pp. 1-24
Author(s):  
VICTORIA SADOVSKAYA
Keyword(s):  
Hyperbolic System ◽  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Riemannian Metrics ◽  
Hölder Continuous ◽  
Group Of Diffeomorphisms ◽  
Small Loss ◽  
Partially Hyperbolic ◽  
Loss Of Regularity

Abstract We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

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