Invariant Measures Satisfying an Equality Relating Entropy,Folding Entropy and Negative Lyapunov Exponents

In this paper we prove that, for a C ² (non-invertible but non-degenerate) map on a compact manifold, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents if and, under a condition on the Jacobian of the map, only if the measure has absolutely continuous conditional measures on the stable manifolds. This work is supported by National Basic Research Program of China (973 Program) (2007CB814800).     

Multifractal Analysis for Lyapunov Exponents on Nonconformal Repellers

For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem nor the classical thermodynamic formalism. We use instead a nonadditive topological pressure to characterize the topological entropy of each level set. This prevents us from estimating the complexity of the level sets using the classical Gibbs measures, which are often one of the main ingredients of multifractal analysis. Instead, we avoid even equilibrium measures, and thus in particular g-measures, by constructing explicitly ergodic measures, although not necessarily invariant, which play the corresponding role in our work.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT by Program POCTI/FEDER and the grant SFRH/BPD/12108/2003.     

Smoothness of Invariant Manifolds for Nonautonomous Equations

For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C^1+α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003.     

Long-Time-Tail Effects on Lyapunov Exponents of a Random, Two-Dimensional Field-Driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, nonoverlapping hard-disk scatterers in a thermostatted electric field, . The low-density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz–Boltzmann equation. In this paper we develop a method to extend theses results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of the Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to nonanalytic, field-dependent contributions to both the positive and negative Lyapunov exponents which are of the form ~ ²ln~, where ~ is a dimensionless parameter proportional to the strength of the applied field. We show that these nonanalytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value due to the presence of the thermostatted field, and that the collision frequency also contains such nonanalytic terms.     

Entropy Production in Nonequilibrium Statistical Mechanics

We consider systems of nonequilibrium statistical mechanics, driven by nonconservative forces and in contact with an ideal thermostat. These are smooth dynamical systems for which one can define natural stationary states μ (SRB in the simplest case) and entropy production e(μ) (minus the sum of the Lyapunov exponents in the simplest case). We give exact and explicit definitions of the entropy production e(μ) for the various situations of physical interest. We prove that e(μ)≥0 and indicate cases where e(μ)>0. The novelty of the approach is that we do not try to compute entropy production directly, but make it depend on the identification of a natural stationary state for the system. Received: 15 July 1996 / Accepted: 30 October 1996     

Using integrability to produce chaos: Billiards with positive entropy

A new open class of convex 2 dimensional planar billiards with positive Lyapunov exponent almost everywhere is constructed. We introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive Lyapunov exponents. We prove that under smallC ⁶ perturbations, focusing arcs remain focusing and thereby show that perturbations of the Bunimovich stadium billiard have positive Lyapunov exponents.Partially supported by NSF grant DMS 8806067     

Ruelle Inequality Relating Entropy,Folding Entropy and Negative Lyapunov Exponents

In this paper we prove an inequality conjectured by Ruelle relating the entropy, folding entropy and negative Lyapunov exponents of a differentiable map on a compact manifold, under a set of conditions on degenerate points of the map.This work is supported by SFMSBRP and NSFDYS     

Lyapunov Instability for a Hard-Disk Fluid in Equilibrium and Nonequilibrium Thermostated by Deterministic Scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.     

Lyapunov exponents and kolmogorov-sinai entropy for a high-dimensional convex billiard

We compute the Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a self-bound N-body system that is realized as a convex billiard. This system exhibits truly high-dimensional chaos, and 2N-4 Lyapunov exponents are found to be positive. The KS entropy increases linearly with the numbers of particles. We examine the chaos generating defocusing mechanism and investigate how high-dimensional chaos develops in this system with no dispersing elements.     

The dynamical entropy of the quantum Arnold cat map

We present a rigorous computation of the dynamical entropyh of the quantum Arnold cat map. This map, which describes a flow on the noncommutative two-dimensional torus, is a simple example of a quantum dynamical system with optimal mixing properties, characterized by Lyapunov exponents ± 1n ⁺, ⁺ > 1. We show that, for all values of the quantum deformation parameter,h coincides with the positive Lyapunov exponent of the dynamics.     

Lyapunov Instability of the Boundary-Driven Chernov–Lebowitz Model for Stationary Shear Flow

We report on the computation of full Lyapunov spectra of the boundary-driven Chernov–Lebowitz model for stationary planar shear flow. The Lyapunov exponents are calculated with a recently developed formalism for systems with elastic hard collisions. Although the Chernov–Lebowitz model is strictly energy conserving, any phase-space volume is subjected to a contraction due to the reflection rules of the hard disks colliding with the walls. Consequently, the sum of Lyapunov exponents is negative. As expected for an inhomogeneously driven system, the Lyapunov spectra do not obey the conjugate pairing rule. The external driving makes the system less chaotic, which is reflected in a decrease of the Kolmogorov–Sinai entropy if the driving is increased.     

Chaos and curvature in a quartic Hamiltonian system

Chaotic behaviour of a quartic oscillator system given byH l/2(p ₁ ² +p ₂ ² )+ (1/12)(1 -α) (q ₁ ⁴ +q ₂ ⁴ )+1/2q ₁ ² q ₂ ² is studied. Though the Riemannian curvature is positive the system is nonintegrable except when S/B α = 0. Calculation of maximal Lyapunov exponents indicates a direct correlation between chaos and negative curvature of the potential boundary.     

Statistical properties of Lyapunov exponents and of quantum conductance of random systems in the regime of hopping transport

The statistics of the zero-temperature conductance and the Lyapunov exponents of one-, two- and three-dimensional disordered systems in the regime of strong localization is studied numerically. In one dimension, the origin of the universality of the moments of the conductance is explained. The relation between the most probable value of the conductance and its configurational average is discussed. The relative fluctuations of the conductance (and of the resistance) are shown to grow exponentially with the system length. In higher dimensions the conductance is almost entirely determined by the smallest of the Lyapunov exponents. The statistics of the conductance is therefore the same as in the one dimensional case. A model is proposed for the treatment of the fluctuations in hopping transport at finite temperatures. An exponential dependence of the relative fluctuations of the conductance/resistance on the temperature is predicted, log (δg/g) ∞ T^?a with α = 1/(d+1). It is concluded that the presently available experimental data on the temperature dependence of the conductance fluctuations in the hopping regime can be understood by replacing the system size in the zerotemperature result for the fluctuations of the conductance by the hopping length.     

Entropy potential and Lyapunov exponents

According to a previous conjecture, spatial and temporal Lyapunov exponents of chaotic extended systems can be obtained from derivatives of a suitable function, the entropy potential. The validity and the consequences of this hypothesis are explored in detail. The numerical investigation of a continuous-time model provides a further confirmation to the existence of the entropy potential. Furthermore, it is shown that the knowledge of the entropy potential allows determining also Lyapunov spectra in general reference frames where the time-like and space-like axes point along generic directions in the space-time plane. Finally, the existence of an entropy potential implies that the integrated density of positive exponents (Kolmogorov-Sinai entropy) is independent of the chosen reference frame. (c) 1997 American Institute of Physics.     

Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos

We analytically link three properties of nonlinear dynamical systems, namely sensitivity to initial conditions, entropy production, and escape rate, in z-logistic maps for both positive and zero Lyapunov exponents. We unify these relations at chaos, where the Lyapunov exponent is positive, and at its onset, where it vanishes. Our result unifies, in particular, two already known cases, namely (i) the standard entropy rate in the presence of escape, valid for exponential functionality rates with strong chaos, and (ii) the Pesin-like identity with no escape, valid for the power-law behavior present at points such as the Feigenbaum one.     

Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms

We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov exponents of the individual members. The motivation for this study is the hope that a rich enough family of diffeomorphisms will always have members with positive Lyapunov exponents, that is to say, positive entropy. At question is what sort of notion of richness would make such a conclusion valid. One type of richness of a family—invariance under the left action of SO(n+1)—occurs naturally in the context of volume preserving diffeomorphisms of the n-sphere. Based on some positive results for families linear maps obtained by Dedieu and Shub, we investigate the exponents of such a family on the 2-sphere. Again motivated by the linear case, we investigate whether there is in fact a lower bound for the mean of the Lyapunov exponents in terms of the random exponents (with respect to the push-forward of Haar measure on SO(3)) in such a family. The family that we study contains a twist map with stretching parameter . In the family , we find strong numerical evidence for the existence of such a lower bound on mean Lyapunov exponents, when the values of the stretching parameter are not too small. Even moderate values of like 10 are enough to have an average of the metric entropy larger than that of the random map. For small the estimated average entropy seems positive but is definitely much less than the one of the random map. The numerical evidence is in favor of the existence of exponentially small lower and upper bounds (in the present example, with an analytic family). Finally, the effect of a small randomization of fixed size of the individual elements of the family is considered. Now the mean of the local random exponents of the family is indeed asymptotic to the random exponent of the entire family as tends to infinity.     

Steady-state electrical conduction in the periodic Lorentz gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a thermostat constructed according to Gauss principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.Dedicated to Elliott Lieb     

Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices

We study the behavior of the generalized Lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsD. For products of random matrices without any particular structure the generalized Lyapunov exponents become equal in this limit and the value of one of the generalized Lyapunov exponents is obtained by simple arguments. On the contrary, for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized Lyapunov exponents remains different forD , indicating that high dimensionality cannot always destroy intermittency.     

Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

We compute the Lyapunov exponent, the generalized Lyapunov exponents, and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, are used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. The Lyapunov exponent for the corresponding map agrees with the conjectured limit _map = -2 log(R) + C + O(R) and we derive an approximate value for the constantC in good agreement with numerical simulations. We also find a diverging diffusion constantD(t)logt and a phase transition for the generalized Lyapunov exponents.