Local rigidity of Lyapunov spectrum for toral automorphisms

Andrey Gogolev; Boris Kalinin; Victoria Sadovskaya

doi:10.1007/s11856-020-2028-6

Local rigidity of higher rank non-abelian action on torus

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.77 ◽

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.

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Local rigidity for certain groups of toral automorphisms

Israel Journal of Mathematics ◽

10.1007/bf02776025 ◽

1991 ◽

Vol 75 (2-3) ◽

pp. 203-241 ◽

Cited By ~ 30

Author(s):

A. Katok ◽

J. Lewis

Keyword(s):

Local Rigidity ◽

Toral Automorphisms

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How Many Inflections are There in the Lyapunov Spectrum?

Communications in Mathematical Physics ◽

10.1007/s00220-021-04161-4 ◽

2021 ◽

Author(s):

O. Jenkinson ◽

M. Pollicott ◽

P. Vytnova

Keyword(s):

Piecewise Linear ◽

Lyapunov Spectrum ◽

Stat Phys ◽

Linear Map ◽

Piecewise Linear Map ◽

Expanding Map ◽

Lyapunov Spectra ◽

Linear Maps ◽

Inflection Points ◽

Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

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Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

International Journal of Mathematics ◽

10.1142/s0129167x15500640 ◽

2015 ◽

Vol 26 (08) ◽

pp. 1550064

Author(s):

Bachir Bekka

Keyword(s):

Von Neumann Algebra ◽

Regular Representation ◽

Von Neumann ◽

Lp Spaces ◽

Local Rigidity ◽

Left Regular Representation ◽

Left Regular ◽

Property T ◽

Finite Factor ◽

Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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A method for validating the accuracy of NMR protein structures

Nature Communications ◽

10.1038/s41467-020-20177-1 ◽

2020 ◽

Vol 11 (1) ◽

Author(s):

Nicholas J. Fowler ◽

Adnan Sljoka ◽

Mike P. Williamson

Keyword(s):

Secondary Structure ◽

Crystal Structures ◽

Protein Structures ◽

Random Coil ◽

Local Rigidity ◽

Correlation Score ◽

Nmr Structures ◽

Explicit Solvent ◽

Structure Ensemble ◽

Better Than

AbstractWe present a method that measures the accuracy of NMR protein structures. It compares random coil index [RCI] against local rigidity predicted by mathematical rigidity theory, calculated from NMR structures [FIRST], using a correlation score (which assesses secondary structure), and an RMSD score (which measures overall rigidity). We test its performance using: structures refined in explicit solvent, which are much better than unrefined structures; decoy structures generated for 89 NMR structures; and conventional predictors of accuracy such as number of restraints per residue, restraint violations, energy of structure, ensemble RMSD, Ramachandran distribution, and clashscore. Restraint violations and RMSD are poor measures of accuracy. Comparisons of NMR to crystal structures show that secondary structure is equally accurate, but crystal structures are typically too rigid in loops, whereas NMR structures are typically too floppy overall. We show that the method is a useful addition to existing measures of accuracy.

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