Partial Hyperbolicity, Lyapunov Exponents and Stable Ergodicity

We show that there exists a C∞ volume preserving topologically transitive diffeomorphism of a compact smooth Riemannian manifold which is ergodic (indeed is Bernoulli) on an open and dense subset \({\mathcal{G}}\) of not full volume and has zero Lyapunov exponent on the complement of \({\mathcal{G}}\) .

We created a triaxial stellar system through the cold dissipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. Once an equilibrium system had been obtained, the multipolar expansion was freezed and smoothed in order to get a stationary smooth potential. The resulting model was self-consistent and the orbits and Lyapunov exponents could then be computed for a randomly selected sample of 3472 of the bodies that make up the system. More than half of the orbits (52.7 % ) turned out to be chaotic. Regular orbits were then classified using the frequency analysis automatic code of Carpintero and Aguilar (1998, MNRAS 298(1), 1–21). We present plots of the distributions of the different kinds of orbits projected on the symmetry planes of the system. We distinguish chaotic orbits with only one non-zero Lyapunov exponent from those with two non-zero exponents and show that their spatial distributions differ, that of the former being more similar to the one of the regular orbits. Most of the regular orbits are boxes and boxlets, but the minor axis tubes play an important role filling in the wasp waists of the boxes and helping to give a lentil shape to the system. We see no problem in building stable triaxial models with substantial amounts of chaotic orbits; the difficulties found by other authors may be due not to a physical cause but to a limitation of Schwarzschild’s method.

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

We show that for any C 1 + α C^{1+\alpha } diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Ω n \Omega _{n} of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence { μ n } \{\mu _{n}\} of f f -invariant ergodic probability measures with s u p p ( μ n ) ⊆ Ω n supp (\mu _{n}) \subseteq \Omega _{n} we have μ n → μ \mu _{n}\to \mu in the weak- ∗ * topology.

—On a short time scale, Atmospheric Angular Momentum (AAM) has been demonstrated to be essentially the sole excitation source of LOD variations. The LOD variation, therefore, merely reflects the AAM variation (LOD as proxy for AAM). The study of the nonlinear nature of AAM variability (e.g., its orbital complexity, dimensionality and extreme sensitivity to the initial conditions) may provide a physical premise for theoretical modelling of the earth-atmosphere-ocean system. Analysis of the high quality of detailed daily LOD/AAM variations time series, spanning the period of 1962–1992, reveals a non-zero and low positive Lyapunov exponent value which suggests possible evidence of deterministic chaos in the underlying dynamics. Application of modern nonlinear prediction techniques capable of distinguishing chaos and random fractals to the data set, further support the above findings and render a predictive time limit of approximately 12–15 days. A low dimensional strange attractor and a low average Lyapunov exponent suggest a low level of unpredictability and stability in the system dynamics. It is argued here that a possible source of the raised entropy in LOD/AAM systems possibly stems from a conceivable nonlinear interaction between the seasonal cycle and inter- or intra-annual fluctuations due to thermodynamics properties of the atmosphere-ocean system.

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We establish stable ergodicity of diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the central direction are all negative with respect to invariant SRB-measures.

We study the hyperbolicity properties of the action of a non-elementary automorphism group on a compact complex surface, with an emphasis on K3 and Enriques surfaces. A first result is that when such a group contains parabolic elements, Zariski diffuse invariant measures automatically have non-zero Lyapunov exponents. In combination with our previous work, this leads to simple criteria for a uniform expansion property on the whole surface, for groups with and without parabolic elements. This, in turn, has strong consequences on the dynamics: description of orbit closures, equidistribution, ergodicity properties, etc. Along the way, we provide a reference discussion on uniform expansion of non-linear discrete group actions on compact (real) manifolds and the construction of Margulis functions under optimal moment conditions.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="PH Subscript mu Superscript 1 Baseline left-parenthesis upper M comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mtext>PH</mml:mtext> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\text {PH}^{1}_\mu (M,3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> conservative partially hyperbolic diffeomorphisms with center dimensions three or less. We prove that there is a dense subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H subset-of PH Subscript mu Superscript 1 Baseline left-parenthesis upper M comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mo> ⊂ </mml:mo> <mml:msubsup> <mml:mtext>PH</mml:mtext> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}\subset \text {PH}^{1}_\mu (M,3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of script upper H"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in \mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has non-zero Lyapunov exponents on a set of positive volume.

Introduction Equivariant geometry and dynamics Technical preliminaries Part 1. Markov Partitions: Markov partitions for finite group actions Transversally hyperbolic sets Markov partitions for basic sets Part 2. Stable Ergodicity: Preliminaries Livsic regularity and ergodic components Stable ergodicity Appendix A. On the absolute continuity of $\nu$ Appendix. Bibliography.

We analyze when partially hyperbolic endomorphisms can be perturbed in order to be close to one with non-zero Lyapunov exponents and with an unique inverse measure. Problems of this nature were already boarded and solved in the setting of diffeomorphisms. The extension to non-invertible maps presents as one the main difficulties the fact of that multivaluated inverse iterations of the map make that the local unstable manifolds may intersect each other since they depend on the whole prehistory.

In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts - julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.

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.

The deterministic equations describing the dynamics of the atmosphere (and of the climate system) are known to display the property of sensitivity to initial conditions. In the ergodic theory of chaos, this property is usually quantified by computing the Lyapunov exponents. In this review, these quantifiers computed in a hierarchy of atmospheric models (coupled or not to an ocean) are analyzed, together with their local counterparts known as the local or finite-time Lyapunov exponents. It is shown in particular that the variability of the local Lyapunov exponents (corresponding to the dominant Lyapunov exponent) decreases when the model resolution increases. The dynamics of (finite-amplitude) initial condition errors in these models is also reviewed, and in general found to display a complicated growth far from the asymptotic estimates provided by the Lyapunov exponents. The implications of these results for operational (high resolution) atmospheric and climate modelling are also discussed.

The objects of the multiplicative ergodic theorem (Lyapunov exponents and Oseledets spaces for products of random matrices) play a key role in the local theory of random dynamical systems. In this paper their behaviour under perturbations is investigated. It is shown that stability of Oseledets spaces is equivalent to stability of Lyapunov exponents.