We exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. This builds on work of Guivarc'h-Raugi and Gol'dsheid-Margulis, who considered products of random matrices, and of Bonatti-Viana, who dealt with the case when the base dynamics is a subshift of finite type. Here the Markov structure may have infinitely many symbols and the ambient space needs not be compact. As an application, in another paper we prove the Zorich-Kontsevich conjecture on the Lyapunov spectrum of the Teichmüller flow in the space of translation surfaces.
We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a u-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched cocycles.
In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmueller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmueller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmueller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech group of these Teichmueller curves. Fina…
We prove the Zorich-Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.
We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $SL_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and…
We prove that the "plus" Rauzy--Veech groups of the connected components of all strata of meromorphic quadratic differentials defined on Riemann surfaces of genus at least one having at most simple poles and at least three singularities (zeros or poles), not all of even order, are finite-index subgroups of their ambient symplectic groups. This shows that the "plus" Lyapunov spectrum of such strata is simple. Moreover, we show that the index of the "minus" Rauzy--Veech group is also finite for connected components of strata satisfying the same conditions and having exactly two singularities of …
We prove that in a $C^1$-open and $C^k$-dense set of some classes of $C^k$ Anosov flows all Lyapunov exponents have multiplicity 1 with respect to appropriate measures. The classes are geodesic flows with equilibrium states of Hölder-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of Hölder-continuous potentials. In the proof, we use and prove perturbative results for jets of flows to modify eigenvalues of certain Poincaré maps and, using a Markov partition, apply the simplicity criterion of Avila and Viana.
We prove that Lyapunov exponents of typical Hölder continuous fiber bunched linear cocycles over singular hyperbolic flows have multiplicity 1: the subspace of exceptional cocycles has infinite codimension.
We study optimization-based criteria for the stability of switching systems, known as Path-Complete Lyapunov Functions, and ask the question "can we decide algorithmically when a criterion is less conservative than another". Our contribution is twofold. First, we show that a Path-Complete Lyapunov Function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each Path-Complete criterion an implied invariant …
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 over hyperbolic systems, such as shift maps or Axiom A diffeomorphisms. In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of $u$-states. Our main …
An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We prove large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we establish continuity properties of the Lyapu…
In the current paper we prove simplicity for the spectrum of Lyapunov exponents for triangle sequence and Selmer algorithm in dimension 3. We introduce a strategy that can be applied for a wide class of Markovian MCF.
A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time switching systems under arbitrary switching. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. The converse, however, is not true even for discrete-time linear systems…
We describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism. The description is expressed in terms of the finest dominated splitting and Lyapunov exponents that appear in the limit of a sequence of periodic orbits, and involves the majorization partial order. Among the applications, we give a simple criterion for the occurrence of universal dynamics.
In the context of mechanical Lagrangian dynamics, we prove a new Lyapunov instability criterion for a non strict local minimum equilibrium point of a smooth potential where the sufficient condition for instability is the existence of a smooth solution of a certain linear PDE derived from the mechanical Lagrangian governing the dynamics. In the presence of a magnetostatic field, we also give an additional sufficient condition for the motion of a charged particle to be Lyapunov unstable.
We study the effects of IID random perturbations of amplitude $ε> 0$ on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a \in [0,1]\}$ of smooth multimodal maps which "predominantly expanding", i.e., $|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0 \}$. We obtain, for any $ε> 0$, a \emph{checkable, finite-time} criterion on the parameter $a$ for random perturbations of the map $f_a$ to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the …
In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the $C^1$ topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.
We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a $C^1$-dense and open subset of the set of a diffeomorphisms having a robust cycle. As a corollary, there exists a $C^1$-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting…
A wide variety of physical systems ranging from the firing of neurons to eutrophication of lakes to the presence of Arctic summer sea ice exhibit a phenomenon known as tipping. In mathematical models, tipping can be caused by bifurcations, noise, and the rate at which parameters are changing in time [2]. Because traditional methods in dynamical systems are usually concerned with the long-term behavior of the system, these methods are not always able to detect the transient dynamics characteristic of rate-tipping. In this paper, we consider one- and two-dimensional dynamical systems with nonaut…
It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n=2\cos\left(\binom{n}{2}ω+ny+x\right)$ with $ω$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $ω$ is the golden mean, which allows to derive the positivity of the infinite-volume Lyapunov exponent from three conditions imposed at a fixed, finite scale. Here we numerically …
Avila and Viana exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. Here, in terms of geometric perturbations, we prove that this sufficient criterion is generic in the space of all fiber bunched linear cocycles over Markov maps: the set of exceptional cocycles has infinite codimention, i.e. it is locally contained in finite unions of closed submanifolds with arbitrarily high codimension.
On the chaotic behavior of the Lagrangian flow of the 2D Navier-Stokes system with bounded degenerate noise
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学-华东师范大学上海数学中心）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
