Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles

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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Random walks in random medium on ? and Lyapunov spectrum

Annales de l Institut Henri Poincaré Probabilités et Statistiques ◽

10.1016/s0246-0203(03)00071-2 ◽

2004 ◽

Vol 40 (3) ◽

pp. 309-336

Author(s):

J Brémont

Keyword(s):

Random Walks ◽

Random Medium ◽

Lyapunov Spectrum

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Lyapunov spectrum of ball quotients with applications to commensurability questions

Duke Mathematical Journal ◽

10.1215/00127094-3165969 ◽

2016 ◽

Vol 165 (1) ◽

pp. 1-66 ◽

Cited By ~ 3

Author(s):

André Kappes ◽

Martin Möller

Keyword(s):

Lyapunov Spectrum ◽

Ball Quotients

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CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407017677 ◽

2007 ◽

Vol 17 (03) ◽

pp. 953-963 ◽

Cited By ~ 4

Author(s):

XIAO-SONG YANG ◽

YAN HUANG

Keyword(s):

Neural Networks ◽

Limit Cycle ◽

Limit Cycles ◽

Cellular Neural Networks ◽

Lyapunov Spectrum ◽

Regular Array ◽

Poincaré Maps ◽

One Dimensional ◽

Poincare Maps ◽

New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

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Simple Lyapunov spectrum for certain linear cocycles over partially hyperbolic maps

Nonlinearity ◽

10.1088/1361-6544/aae939 ◽

2018 ◽

Vol 32 (1) ◽

pp. 238-284

Author(s):

Mauricio Poletti ◽

Marcelo Viana

Keyword(s):

Lyapunov Spectrum ◽

Partially Hyperbolic

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Symmetry of attractors and the Perron-Frobenius operator

The Journal of Difference Equations and Applications ◽

10.1080/10236190601045788 ◽

2006 ◽

Vol 12 (11) ◽

pp. 1147-1178 ◽

Cited By ~ 2

Author(s):

Prashant G. Mehta ◽

Mirko Hessel-von Molo ◽

Michael Dellnitz

Keyword(s):

Frobenius Operator

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On the convergence to equilibrium states for certain non-hyperbolic systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385797086240 ◽

1997 ◽

Vol 17 (4) ◽

pp. 977-1000 ◽

Cited By ~ 14

Author(s):

MICHIKO YURI

Keyword(s):

Periodic Orbits ◽

Invariant Measures ◽

Hyperbolic Systems ◽

Markov Property ◽

Standard Technique ◽

Equilibrium States ◽

Convergence To Equilibrium ◽

Invariant Density ◽

Frobenius Operator ◽

Uniformly Bounded

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

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