Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise

We consider the stochastic Ginzburg–Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg–Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for thelow-lying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Received: 10 September 2000 / Accepted: 13 December 2000     

On “Thermodynamics” of Rational MapsI.¶Negative Spectrum

We study the pressure spectrum P(t) of the maximal measure for arbitrary rational maps. We also consider its modified version which is defined by means of the variational principle with respect to non-atomic invariant measures. It is shown that for negative values of t, the modified spectrum has all major features of the hyperbolic case (analyticity, the existence of a spectral gap for the corresponding transfer operator, rigidity properties, etc). The spectrum P(t) can be computed in terms of . Their Legendre transforms are the Hausdorff and the box-counting dimension spectra of the maximal measure respectively. This work is closely related to a paper [32] by D. Ruelle. Received: 2 August 1999 / Accepted: 11 January 2000     

Asymptotic Behavior of Thermal Nonequilibrium Steady States for a Driven Chain of Anharmonic Oscillators

We consider a model of heat conduction introduced in [6], which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures. We study the low temperature asymptotic behavior of the invariant measure. We show that, in this limit, the invariant measure is characterized by a variational principle. The main technical ingredients are some control theoretic arguments to extend the Freidlin–Wentzell theory of large deviations to a class of degenerate diffusions. Received: 6 January 2000 / Accepted: 4 May 2000     

Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings

We first study a family of invariant transformations for the integer moment problem. The fixed point of these transformations generates a positive measure with support on a Cantor set depending on a parameter q. We analyze the structure and properties of the set of orthogonal polynomials with respect to this measure. Among these polynomials, we find the iterates of the canonical quadratic mapping: F(x)=(x–q) ², q2. It appears that the measure is invariant with respect to this mapping. Algebraic relations among these polynomials are shown to be analytically continuable below q=2, where bifurcation doubling among stable cycles occurs. As the simplest possible consequence we analyze the neighborhood of q=2 (transition region) for q<2.     

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs. It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type. For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1. We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. Received: 27 March 2000 / Accepted: 7 December 2000     

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.     

Conformally Invariant Powers of the Laplacian, <Emphasis Type="Italic">Q</Emphasis>-Curvature,and Tractor Calculus

 We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6 and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on [12], where the relationship of the normal standard tractor bundle to the ambient construction is described. Received: 24 January 2002 / Accepted: 1 November 2002 Published online: 18 February 2003 Communicated by P. Sarnak     

The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson–Walker Spacetimes

We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on Robertson–Walker spacetimes. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. It is demonstrated that any two such states (of sufficiently high order) are locally quasi-equivalent. We give a microlocal characterisation of spinor Hadamard states and we show that this agrees with the usual characterisation of such states in terms of the singular behaviour of their associated twopoint functions. The polarisation set of these twopoint functions is determined and found to have a natural geometric form. We finally prove that our adiabatic states of infinite order are Hadamard, and that those of order n correspond, in some sense, to a truncated Hadamard series and therefore allow for a point splitting renormalisation of the expected stress-energy tensor. Received: 30 June 1999 / Accepted: 21 September 2000     

Constant-Length Random Substitutions and Gibbs Measures

This work is devoted to the study of processes generated by random substitutions over a finite alphabet. We prove, under mild conditions on the substitution’s rule, the existence of a unique process which remains invariant under the substitution, and which exhibits a polynomial decay of correlations. For constant-length substitutions, we go further by proving that the invariant state is precisely a Gibbs measure which can be obtained as the projective limit of its natural Markovian approximations. We end up the paper by studying a class of substitutions whose invariant state is the unique Gibbs measure for a hierarchical two-body interaction.     

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     

Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation

We develop a new analysis of the order-disorder transition in ferromagnetic Potts models for large numberq of spin states. We use the Pirogov-Sinaï theory which we adapt to the Fortuin-Kasteleyn representation of the models. This theory applies in a rather direct way in our approach and leads to a system of non-interacting contours with small activities. As a consequence, simpler and more natural techniques are found, allowing us to recover previous results on the bulk properties of the model (which then extend to non-integer values ofq) and to deal with non-translation invariant boundary conditions. This will be applied in a second part of this work to study the behaviour of the interfaces at the transition point.Laboratoire Propre du CNRS: LP 7061     

Motion by Mean Curvature from the Ginzburg-Landau Interface Model

We consider the scalar field φ _t with a reversible stochastic dynamics which is defined by the standard Dirichlet form relative to the Gibbs measure with formal energy . The potential V is even and strictly convex. We prove that under a suitable large scale limit the φ _t -field becomes deterministic such that locally its normal velocity is proportional to its mean curvature, except for some anisotropy effects. As an essential input we prove that for every tilt there is a unique shift invariant, ergodic Gibbs measure for the -field. Received: 1 February 1996 / Accepted: 2 July 1996     

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     

Statistical mechanics of a one-dimensional lattice gas

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be aK-system.     

Ergodicity of the 2D Navier--Stokes Equations¶with Random Forcing

We consider the Navier–Stokes equation on a two dimensional torus with a random force, acting at discrete times and analytic in space, for arbitrarily small viscosity coefficient. We prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time. Received: 18 May 2000 / Accepted: 8 December 2000     

Symmetry and Resonance in Periodic FPU Chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This “Birkhoff–Gustavson normal form” retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems. So the periodic FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries. Received: 25 July 2000 / Accepted: 20 December 2000     

Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional phase space

When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We develop a new robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithm reported here is developed in the exterior algebra representation, which leads to a fast algorithm with some novel properties. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg–de Vries equation, and the long-wave–short-wave resonance equations are presented. Part 1 considers the case of a four-dimensional phase space, and Part 2 considers the case of a 2n-dimensional phase space with n>2.