Related papers: Random 2D linear cocycles II: statistical properti…
A deep analysis of the Lyapunov exponents, for stationary sequence of matrices going back to Furstenberg, for more general linear cocycles by Ledrappier and generalized to the context of non-linear cocycles by Avila and Viana, gives an…
The paper is concerned with random $S$-adic systems arising from an i.i.d. sequence of unimodular substitutions. Using equidistribution results of Benoist and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the…
We prove measurable Livsic theorems for dynamical systems modelled by Markov Towers. Our regularity results apply to solutions of cohomological equations posed on Henon-like mappings and a wide variety of nonuniformly hyperbolic systems. We…
The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work on the linear stability with constant coefficients, the problem has a free boundary…
We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion…
We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes…
We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic $\mathrm{SL}(2,\mathbb R)$-cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space $G^2$ is the transition space of…
We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of…
In the present paper we give a positive answer to some questions posed by Viana on the existence of positive Lyapunov exponents for Hamiltonian linear differential systems. We prove that there exists an open and dense set of Hamiltonian…
We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is…
A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions it is known to have a unique…
We study the Lyapunov instability of a two-dimensional fluid composed of rigid diatomic molecules, with two interaction sites each, and interacting with a WCA site-site potential. We compute full spectra of Lyapunov exponents for such a…
In this paper we first obtain a formula of averaged Lyapunov exponents for ergodic Szego cocycles via the Herman-Avila-Bochi formula. Then using acceleration, we construct a class of analytic quasi-periodic Szego cocycles with uniformly…
We study random products of matrices in SL_2(C) from the point of view of holomorphic dynamics. For non-elementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a…
We prove the H\"older continuity of Lyapunov exponents for general linear cocycles when the base measures vary in Wasserstein distance, under the assumption of uniform large deviations type (LDT) estimates. This is a measure version of the…
Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by Furstenberg, Guivarc'h-Raugi, Gol'dsheid-Margulis and, more recently, Bonatti-Viana and Avila-Viana. In all the cases, the authors consider cocycles…
In both the random hopping model and at topological phase transitions in one-dimensional chiral systems, the Lyapunov exponent vanishes at zero energy, but is here shown to have an inverse logarithmic increase with a coefficient that is…
For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's…
For substitution systems and translation flows, a new cocycle, which we call {\em spectral cocycle}, is introduced, whose Lyapunov exponents govern the local dimension of the spectral measure for higher-level cylindrical functions. The…
We prove that infinitely renormalizable contracting Lorenz maps with bounded geometry or the so-called {\it a priori bounds} satisfies the slow recurrence condition to the singular point $c$ at its two critical values $c_1^-$ and $c_1^+$.…