Positive Lyapunov exponents for a dense set of bounded measurable SL(2, )-cocycles (original) (raw)
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The upper Lyapunov exponent of Sl(2,R) cocycles: Discontinuity and the problem of positivity
Lecture Notes in Mathematics, 1991
Let T be an aperiodic automorphism of a standard probability space (X,m). Let V be the subset of A = L°°(X% 5/(2, R)) where the upper Lyapunov exponent is positive almost everywhere. We prove that the set V \ int(V) is not empty. So, there are always points in A where the Lyapunov exponents are discontinuous. We show further that the decision whether a given cocycle is in V is at least as hard as the following cohomology problem: Can a given measurable set Z C X be represented as YAT(Y) for a measurable set Y C X?
Density of positive Lyapunov exponents for quasiperiodic SL(2, R) -cocycles in arbitrary dimension
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We show that given a fixed irrational rotation of the d-dimensional torus, any analytic SL(2, R) cocycle can be perturbed so that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [K] and Fayad-Krikorian [FK]. The key technique is the analiticity of m-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten Martini Problem [AJ]. In the appendix, we discuss the smoothness of m-functions for a larger class of systems including the skew-shift.
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We study the positivity of the Lyapunov exponent for a smooth SL(2, R)valued cocycle defined over a flow from a class which includes the Kronecker flows and others as well. We also discuss the question of the density in the Hölder class of the set of SL(2, R)-cocycles exhibiting an exponential dichotomy when the base flow is of Kronecker type on a 2-torus.
Nonexistence of Lyapunov exponents for matrix cocycles
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2017
It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f : X → X with exponential specification property and a Hölder continuous matrix cocycle A : X → G(m, R), we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A is residual (i.e., containing a dense G δ set).
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Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.
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The celebrated Oseledets theorem \cite{O}, building over seminal works of Furstenberg and Kesten on random products of matrices and random variables taking values on non-compact semisimple Lie groups \cite{FK,Furstenberg}, ensures that the Lyapunov exponents of mathrmSL(d,mathbbR)\mathrm{SL}(d,\mathbb R)mathrmSL(d,mathbbR)-cocycles (dgeqslant2)(d\geqslant 2)(dgeqslant2) over the shift are well defined for all points in a total probability set, ie, a full measure subset for all invariant probabilities. Given a locally constant mathrmSL(d,mathbbR)\mathrm{SL}(d,\mathbb R)mathrmSL(d,mathbbR)-valued cocycle we are interested both in the set of points on the shift space for which some Lyapunov exponent is not well defined, and in the set of directions on the projective space mathbfPmathbbRd\mathbf P \mathbb R^dmathbfPmathbbRd along which there exists no well defined exponential growth rate of vectors for a certain product of matrices. We prove that if the semigroup generated by finitely many matrices in mathrmSL(d,mathbbR)\mathrm{SL}(d,\mathbb R)mathrmSL(d,mathbbR) is not compact and is strongly projectively accessible then there exists a dens...
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We provide an example of a Schrödinger cocycle over a mixing Markov shift for which the integrated density of states has a very weak modulus of continuity, close to the log-Hölder lower bound established by W. Craig and B. Simon in [6]. This model is based upon a classical example due to Y. Kifer [15] of a random Bernoulli cocycle with zero Lyapunov exponents which is not strongly irreducible. It follows that the Lyapunov exponent of a Bernoulli cocycle near this Kifer example cannot be Hölder or weak-Hölder continuous, thus providing a limitation on the modulus of continuity of the Lyapunov exponent of random cocycles.
Journal of Dynamics and Differential Equations, 2011
Consider the class of C r-smooth SL(2, R) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: α − pn qn ≤ Cexp(−q r+1+κ n), where C > 0 and 0 < κ < 1 are some constants and Pn qn is some sequence of irreducible fractions.
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arXiv (Cornell University), 2021
JAMERSON BEZERRA, ADRIANA SÁNCHEZ, AND EL HADJI YAYA TALL A. We show that the top Lyapunov exponent + () , = (1 ,. .. ,) with > 0 for each , associated with a random product of quasi-periodic cocycles depends real analytically on the transition probabilities whenever + () is simple. Moreover if the spectrum at is simple (all Lyapunov exponents having multiplicity one) then all Lyapunov exponents depend real analytically on .