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).     

The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators

This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas.     

Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems

This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the nth iteration of a Poincaré map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an epsilon neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail.     

Iterated function systems and dynamical systems

We study the relationship between measures invariant for a piecewise expanding transformation tau of a compact metric space endowed with a underlying measure and measures invariant for an iterated function system T(tau), generated by inverse branches of tau. The main result says that the tau-invariant absolutely continuous measure &mgr; is also T(tau) invariant if and only if tau is absolutely continuously conjugated with a piecewise linear transformation. Measures of maximal entropy and general equilibrium states are also discussed. (c) 1995 American Institute of Physics.     

Stochastic Stability¶for Contracting Lorenz Maps and Flows

In a previous work [M], we proved the existence of absolutely continuous invariant measures for contracting Lorenz-like maps, and constructed Sinai–Ruelle–Bowen measures f or the flows that generate them. Here, we prove stochastic stability for such one-dimensional maps and use this result to prove that the corresponding flows generating these maps are stochastically stable under small diffusion-type perturbations, even though, as shown by Rovella [Ro], they are persistent only in a measure theoretical sense in a parameter space. For the one-dimensional maps we also prove strong stochastic stability in the sense of Baladi and Viana[BV]. Received: 24 February 1999 / Accepted: 7 January 2000     

Natural Equilibrium States for Multimodal Maps

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −t log |Df|, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.     

On a normal form of symmetric maps of [0, 1]

A class of continuous symmetric mappings of [0, 1] into itself is considered leaving invariant a measure absolutely continuous with respect to the Lebesgue measure.     

An analytical construction of the SRB measures for Baker-type maps

For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure (SRB measure) weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is nonconservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction R<-->I<-->P. For the second example, the cases with and without escape are considered. For the last example, we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rham's functional equation and, employing this analogy, we explicitly construct the SRB measures. (c) 1998 American Institute of Physics.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Computing Invariant Densities and Metric Entropy

We present a method for accurately computing the metric entropy (or, equivalently, the Lyapunov exponent) of the absolutely continuous invariant measure μ for a piecewise analytic expanding Markov map T of the interval. We construct atomic signed measures μ _M supported on periodic orbits up to period M, and prove that super-exponentially fast. We illustrate our method with several examples. Received: 25 July 1999 / Accepted: 7 January 2000     

Information flow and maximum entropy measures for 1-D maps

The invariant measures of maximal metric entropy are constructed explicitly for some maps of the interval, by iterating the maps backward. The construction illustrates in a particularly clear way the information flow in simple systems, as well as recently conjectured relationships between dimensions of invariant measures, Lyapunov exponents, and entropies. maps, it is conjectured that the natural measure is the invariant measure with strongest mixing.     

Resonance Expansions in Semi-Classical Propagation

We consider SRB-measures of coupled map lattices. The emphasis is given to a definition according to which a SRB-measure is an invariant probability measure whose projections onto finite-dimensional subsystems are absolutely continuous with respect to the Lebesgue measure. We show that coupled map lattices which are close to an uncoupled expanding map have typically an infinite number of SRB-measures. In particular, we give a counterexample to the Bricmont–Kupiainen conjecture. Received: 23 June 2000 / Accepted: 4 January 2001     

Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.     

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.     

On YM2 measures and area-preserving diffeomorphisms

For a given gauge group and compact Riemannian two-manifold, it is known that the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all area-preserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiff-equivariant equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the Yang-Mills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyze the action of SDiff.     

Non-lacunary Gibbs Measures for Certain Fractal Repellers

In this paper, we study non-uniformly expanding repellers constructed as the limit sets for a non-uniformly expanding dynamical systems. We prove that given a Hölder continuous potential φ satisfying a summability condition, there exists non-lacunary Gibbs measure for φ, with positive Lyapunov exponents and infinitely many hyperbolic times almost everywhere. Moreover, this non-lacunary Gibbs measure is an equilibrium measure for φ.     

On noninvertible mappings of the plane: Eruptions

In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption," is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve. (c) 1996 American Institute of Physics.     

Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so-called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking.     

Phase Transition and Semi-Global Reducibility

For 1D continuous Schödinger operators with large analytic quasi-periodic potentials of two frequencies, one knows that the spectral measure is singular at the bottom of the spectrum and purely absolutely continuous in the upper part of the spectrum, so there is a phase transition when energy increases. In this paper, we obtain the exact power-law for the phase transition in energy by the semi-global reducibility theory of analytic quasi-periodic linear systems.     

Anomalous Dissipation in a Stochastically Forced Infinite-Dimensional System of Coupled Oscillators

We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed.