On the Lyapunov spectrum of relative transfer operators

We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the [Formula: see text]-topology by positive matrices with an associated dominated splitting.

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Totally Positive Matrices

10.1017/cbo9780511691713 ◽

2009 ◽

Cited By ~ 16

Author(s):

Allan Pinkus

Keyword(s):

Totally Positive ◽

Positive Matrices ◽

Totally Positive Matrices

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Limit set trichotomy and dichotomy for skew-product semiflows with applications

Nonlinear Analysis ◽

10.1016/j.na.2009.01.135 ◽

2009 ◽

Vol 71 (7-8) ◽

pp. 2834-2839

Author(s):

Bin-Guo Wang ◽

Wan-Tong Li

Keyword(s):

Limit Set ◽

Skew Product ◽

Limit Set Trichotomy

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On Gibbs Measures and Spectra of Ruelle Transfer Operators

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-073-2 ◽

2017 ◽

Vol 60 (2) ◽

pp. 411-421

Author(s):

Luchezar Stoyanov

Keyword(s):

Spectral Radius ◽

Spectral Properties ◽

Gibbs Measures ◽

Transfer Operator ◽

Frobenius Theorem ◽

Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

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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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Vortices over Riemann surfaces and dominated splittings

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.142 ◽

2021 ◽

pp. 1-26

Author(s):

THOMAS METTLER ◽

GABRIEL P. PATERNAIN

Keyword(s):

Euler Characteristic ◽

Tangent Bundle ◽

Riemann Surfaces ◽

Dominated Splitting ◽

Vortex Equations ◽

Special Cases ◽

Unit Tangent Bundle ◽

Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

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