Singular analytic linear cocycles with negative infinite Lyapunov exponents (original) (raw)

Remarks concerning the Lyapunov exponents of linear cocycles

We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocycle over a compact metric minimal flow. This condition turns out to have significant consequences for the dynamics of the cocycle. We make use of such classical ODE techniques as the Lyapunov-Perron triangularization method, and the ergodic-theoretical techniques of Krylov and Bogoliubov.

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).

Complex one-frequency cocycles

Journal of the European Mathematical Society, 2014

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.

On the Lyapunov exponent of certain SL(2,ℝ)-valued cocycles II

Differential Equations and Dynamical Systems, 2010

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.

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Communications in Mathematical Physics, 2014

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.

A generic bounded linear cocycle has simple Lyapunov spectrum

Ergodic Theory and Dynamical Systems, 2005

We show that the set of cocycles with integral separateness is open and dense in the space of all bounded Gl(d, R)-cocycles equipped with uniform topology. As a consequence, a generic bounded linear cocycle has simple Lyapunov spectrum and dominated Oseledets splitting, and a generic bounded Sl(2, R)-cocycle is uniformly hyperbolic.

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?

Lyapunov non-typical behavior for linear cocycles through the lens of semigroup actions

2021

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...

Density of positive Lyapunov exponents for quasiperiodic SL(2, R) -cocycles in arbitrary dimension

Journal of Modern Dynamics, 2010

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.