Multifractal Formalism for Lyapunov Exponents of Fiber-Bunched Linear Cocycles

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Hölder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters. This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.

An advanced combined quantum-dynamic and chaos-geometric method for analysis, modelling, and forecasting of the chaotic dynamics of diatomic molecules in an intense electromagnetic field is presented. The method is based on the use of the non-stationary theory of the Schrödinger equation in the approximation of the density functional and the methods of the theory of chaos and dynamic systems for the analysis of time series of polarization and other characteristics of diatomic molecules in an intense electromagnetic field. In particular, the latter includes the Gottwald-Melbourne test, the correlation integral method , fractal and multifractal formalism, average mutual information, false nearest neighbours, surrogate data algorithms, analysis on the basis of the Lyapunov's exponents, Kolmogorov entropy, nonlinear forecast models based on algorithms of optimized predicted trajectories, B-spline approximations. As an illustration, the advanced data for the dynamical and topological invariants (correlation dimension, embedding dimension, Kaplan-York dimension, Lyapunov's exponents, Kolmogorov entropy, etc.) for the diatomic ZrO molecule in a linearly polarized electromagnetic field are listed.

We prove that any uniformly exponentially stable linear cocycle of matrices defined over a topological dynamical system can be reduced via suitable change of variables to a linear cocycle whose generator has a spectral norm less than 1 at each point. Moreover, we establish an analogous result for nonuniformly exponentially stable linear cocycles, i.e. cocycles with negative Lyapunov exponents. In addition, by using tools from ergodic theory, we obtain converse results that give conditions for (non)uniform exponential stability of linear cocyles under the assumption that they admit a suitable reduction to a linear cocycle whose generator has a spectral norm less than 1 at points that form a ‘large’ subset of our topological dynamical system in the appropriate sense. Finally, we discuss the versions of our results in the case of (non)uniformly hyperbolic linear cocyles.

We study a linear cocycle over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed the cocycle is generated by a $C^{1}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Schr\"odinger equation. Under assumption the eigenvalues of $A_{\varepsilon}(x)$ to be of the form $\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter $\varepsilon$. We show that in the limit $\varepsilon\to 0$ the cocycle "typically" exhibits ED only if it is exponentially close to a constant cocycle. In contrary, if the cocycle is not close to a constant one it does not posesses ED, whereas the Lyapunov exponent is "typically" large.

We prove Hölder continuity of Lyapunov exponents for general linear cocycles varying measures on the base in Wasserstein distance under the assumption of uniform large deviations type (LDT) estimates. This is a measure version of the abstract continuity theorem (ACT) established by Duarte-Klein [23]. The main obstacle here lies in the fact that the magnitude of the exceptional sets in LDT estimates is constantly changing when the base measures deviate. We overcome this via a combination of a Urysohn-type lemma and properties of Wasserstein distance in every iteration step. Our measure version of ACT, combined with the original work of Duarte-Klein [23], provides a complete scheme for proving joint Hölder continuity of Lyapunov exponents with respect to both measure and fiber which resolves all parameter dependence. This continuity theorem is general and applicable to a wide range of mathematical models, including product of random matrices and cocycles essentially generated by shifts. In particular, it applies to associated Schrödinger operators which are central objects in the study of mathematical physics.

Mixing (stretching and folding of fluid elements) in chaotic flows is an iterative process generating presumably self-similar distributions of stretching and striation thickness. This hypothesis is investigated using scaling and multifractal techniques for two prototypical time-periodic chaotic flows: one containing no detectable islands (egg-beater flow), the other involving sizable islands as well as no-slip boundaries (flow between eccentric cylinders). The simplest picture arises in the egg-beater flow. Stretching is well described by multifractal scaling if the very high tail of the distribution of stretchings is neglected. Different methods for obtaining the spectrum of fractal dimensions f(α) agree reasonably well, producing a time-independent self-similar distribution. On the other hand, in the flow between eccentric cylinders, the negative moments do not scale, and the spectrum f(α) is time-dependent (and therefore, it is not self-similar). Due to the extremely wide range of values of stretching, a very large number of points needs to be considered in order to characterize mixing in chaotic systems using a multifractal formalism; this suggests that more work is needed in order to understand finite-size effects and how asymptotic states are reached. However, for cases where multifractal scaling applies, it is possible to relate coarse-grained variables (e.g., intermaterial area density) to microscopic features of the flow (e.g., finite-time Lyapunov exponents).

ABSTRACTWe give a characterization of the notion of a tempered exponential dichotomy on a Banach space in terms of an admissibility property. We note that for a linear cocycle over a measure-preserving transformation satisfying a certain integrability assumption, it follows from the multiplicative ergodic theorem that the dynamics admits a tempered exponential dichotomy if and only if all Lyapunov exponents are non-zero almost everywhere. As a consequence of our approach, we give a new proof of the robustness property of the notion of a tempered exponential dichotomy under sufficiently small linear perturbations and we establish a version of the Grobman–Hartman theorem yielding the existence of topological conjugacies between a linear dynamics with a tempered exponential dichotomy and any sufficiently small nonlinear perturbation. In addition, we show that the conjugacy maps vary continuously with the perturbation.

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 Holder 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.

We consider Hölder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer’s Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the norm and the quasiconformal distortion from the periodic data.

We consider $ \mathit{Sp}\left( 2\mathit{d},\mathbb{R} \right)$ cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology.

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber-bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.