Ruelle's Theorem Research Articles

The Lanford–Ruelle theorem for actions of sofic groups

Let Γ \Gamma be a sofic group, Σ \Sigma be a sofic approximation sequence of Γ \Gamma and X X be a Γ \Gamma -subshift with non-negative sofic topological entropy with respect to Σ \Sigma . Further assume that X X is a shift of finite type, or more generally, that X X satisfies the topological Markov property. We show that for any sufficiently regular potential f : X → R f \colon X \to \mathbb {R} , any translation-invariant Borel probability measure on X X which maximizes the measure-theoretic sofic pressure of f f with respect to Σ \Sigma is a Gibbs state with respect to f f . This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic. As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. On the expository side, we present a short proof of Chung’s variational principle for sofic topological pressure.

Thermodynamic formalism for quantum channels: Entropy, pressure, Gibbs channels and generic properties

Denote [Formula: see text] the set of complex [Formula: see text] by [Formula: see text] matrices. We will analyze here quantum channels [Formula: see text] of the following kind: given a measurable function [Formula: see text] and the measure [Formula: see text] on [Formula: see text] we define the linear operator [Formula: see text], via the expression [Formula: see text]. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where [Formula: see text] was the identity. Under some mild assumptions on the quantum channel [Formula: see text] we analyze the eigenvalue property for [Formula: see text] and we define entropy for such channel. For a fixed [Formula: see text] (the a priori measure) and for a given a Hamiltonian [Formula: see text] we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such [Formula: see text]) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed [Formula: see text] (with more than one point in the support) the set of [Formula: see text] such that it is [Formula: see text]-Erg (also irreducible) for [Formula: see text] is a generic set. We describe a related process [Formula: see text], [Formula: see text], taking values on the projective space [Formula: see text] and analyze the question of the existence of invariant probabilities. We also consider an associated process [Formula: see text], [Formula: see text], with values on [Formula: see text] ([Formula: see text] is the set of density operators). Via the barycenter, we associate the invariant probability mentioned above with the density operator fixed for [Formula: see text].

Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups

We formulate and prove a very general relative version of the Dobrushin–Lanford–Ruelle theorem which gives conditions on constraints of configuration spaces over a finite alphabet such that for every absolutely summable relative interaction, every translation-invariant relative Gibbs measure is a relative equilibrium measure and vice versa. Neither implication is true without some assumption on the space of configurations. We note that the usual finite type condition can be relaxed to a much more general class of constraints. By ‘relative’ we mean that both the interaction and the set of allowed configurations are determined by a random environment. The result includes many special cases that are well known. We give several applications including (1) Gibbsian properties of measures that maximize pressure among all those that project to a given measure via a topological factor map from one symbolic system to another; (2) Gibbsian properties of equilibrium measures for group shifts defined on arbitrary countable amenable groups; (3) A Gibbsian characterization of equilibrium measures in terms of equilibrium condition on lattice slices rather than on finite sets; (4) A relative extension of a theorem of Meyerovitch, who proved a version of the Lanford–Ruelle theorem which shows that every equilibrium measure on an arbitrary subshift satisfies a Gibbsian property on interchangeable patterns.

Gibbs and equilibrium measures for some families of subshifts

AbstractFor subshifts of finite type (SFTs), any equilibrium measure is Gibbs, as long as$f$has$d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobrušin states that for strongly irreducible subshifts, shift-invariant Gibbs measures are equilibrium measures. Here we prove a generalization of the Lanford–Ruelle theorem: for all subshifts, any equilibrium measure for a function with$d$-summable variation is ‘topologically Gibbs’. This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts:$\beta $-shifts, Dyck shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford–Ruelle-type theorem holds. For each of these families, we provide a specific proof of the result.

A Ruelle Operator for continuous time Markov Chains

We consider a generalization of the Ruelle theorem for thecase of continuous time problems. We present a result which we believeis important for future use in problems in Mathematical Physics relatedto C-Algebras.We consider a nite state set S and a stationary continuous timeMarkov Chain Xt, t 0, taking values on S. We denote by the setof paths w taking values on S (the elements w are locally constant withleft and right limits and are also right continuous on t). We consideran innitesimal generator L and a stationary vector p0. We denoteby P the associated probability on (; B). All functions f we considerbellow are in the set L1(P).From the probability P we dene a Ruelle operator Lt; t 0, actingon functions f : ! R of L1(P). Given V : ! R, such that isconstant in sets of the form fX0 = cg, we dene a modied Ruelleoperator ~ LtV ; t 0, in the following way: there exist a certain fV suchthat for each t we consider the operator acting on g given by~ LtV (g)(w) = [1fVLt(eR t0 (V s)(:)ds g fV ) ] (w)We are able to show the existence of an eigenfunction u and aneigen-probability V on associated to ~ LtV ; t 0.We also show the following property for the probability V : for anyintegrable g 2 L1(P) and any real and positive tZe

A Ruelle Operator for continuous time Markov Chains

We consider a generalization of the Ruelle theorem for thecase of continuous time problems. We present a result which we believeis important for future use in problems in Mathematical Physics relatedto C-Algebras.We consider a nite state set S and a stationary continuous timeMarkov Chain Xt, t 0, taking values on S. We denote by the setof paths w taking values on S (the elements w are locally constant withleft and right limits and are also right continuous on t). We consideran innitesimal generator L and a stationary vector p0. We denoteby P the associated probability on (; B). All functions f we considerbellow are in the set L1(P).From the probability P we dene a Ruelle operator Lt; t 0, actingon functions f : ! R of L1(P). Given V : ! R, such that isconstant in sets of the form fX0 = cg, we dene a modied Ruelleoperator ~ LtV ; t 0, in the following way: there exist a certain fV suchthat for each t we consider the operator acting on g given by~ LtV (g)(w) = [1fVLt(eR t0 (V s)(:)ds g fV ) ] (w)We are able to show the existence of an eigenfunction u and aneigen-probability V on associated to ~ LtV ; t 0.We also show the following property for the probability V : for anyintegrable g 2 L1(P) and any real and positive tZe

Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials

We are going to prove a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron–Frobenius–Ruelle operators associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. It is obtained using some sufficient conditions for two-weight boundedness of the Hilbert transform. It has been proved in [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2–3) (2006) 215–230] for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t = 0 that the Lipschitz estimate becomes exponentially better when the dimension of the Jacobi matrix grows. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. We suggest a scheme how to approach Bellissard's problem for all hyperbolic dynamics by uniting the methods of the present paper and those of [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2–3) (2006) 215–230]. On the other hand, the nearness of Jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures. One can notice that this last claim just gives us the classical commutative Perron–Frobenius–Ruelle theorem (it is concerned exactly with the nearness of such measures). In particular, in many situations we can see that the classical Perron–Frobenius–Ruelle theorem is a corollary of a certain non-commutative observation concerning the quantitative nearness of pertinent Jacobi matrices in operator norm.

The Radon–Nikodym problem for approximately proper equivalence relations

We study the Radon–Nikodym problem for approximately proper equivalence relations and more specifically the uniqueness of certain Gibbs states. One of our tools is a variant of the dimension group introduced in the study of approximately finite-dimensional (AF) algebras. As applications, we retrieve sufficient conditions for the uniqueness of traces on AF algebras and parts of the Perron–Frobenius–Ruelle theorem.

Riesz–Kolmogorov Compactness Criterion, Lorentz Convergence and Ruelle Theorem on Locally Compact Abelian Groups

Ruelle's theorem gives, for a certain class of self-adjoint operators on L2(R n ), a description of the pure point and continuous spectral subspaces of the operator in terms of bound and scattering states. We extend this characterization to arbitrary self-adjoint operators acting in L2(X), where X is an Abelian locally compact group. We replace the convergence in Cesaro mean from the standard version of Ruelle's theorem by convergence in Lorentz sense, which is sharper than any convergence in invariant mean sense. Our main tool is a description in term of position and momentum observables of relatively compact subsets of L2(X) extending the Riesz–Kolmogorov theorem.

On Ruelle-Perron-Frobenius Operators.¶I. Ruelle Theorem

We study Ruelle–Perron–Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper, we give a proof of the Ruelle Theorem on Gibbs measures. It is the first part of our research on the subject. The rate of convergence of powers of the operator will be presented in a forthcoming paper.

Sur un théorème spectral et son application aux noyaux lipchitziens

The aim of this note is to point out that the Ionescu Tulcea and Marinescu theorem can be reinforced, using a Nussbaum formula for the essential spectral radius of an operator. In this stronger version, this theorem is suitable for the spectral analysis of lipschitzian, positive, not necessarily markovian kernels, Ruelle theorem follows. As an application to Probability Theory, a large deviation theorem is proved.

A short proof of a theorem of Ruelle

Using the existence of conformal measures in the sense ofSullivan [4] andDenker, Urbanski [2] we give a simple proof of Ruelle's Theorem, that the Hausdorff dimension of a hyperbolic Julia set is the solution of the equationP(R,−tlog|R′|)=0.

A uniqueness theorem for systems of interacting polymers

In this paper we study analyticity properties of the pressure of general polymer systems by application of Ruelle's theorem on the zeros of Asano contracted polynomials to the partition function. Then we prove a uniqueness theorem for the equilibrium state of the investigated class of classical particle systems for a variety of finite range interactions.

A generalization of Ruelle's theorem

A generalization of Ruelle's theorem

Ruelle's theorem and the theory of quasianalytic classes of functions

Ruelle's theorem and the theory of quasianalytic classes of functions

Ferromagnetic spin systems at low temperatures

Finite-spin systems with ferromagnetic, finite range interactions are considered. Using Ruelle's theorem on zeros of polynomials contracted according to Asano, analyticity of pressure and correlation functions is proved. A description of all translation invariant equilibrium states at low temperatures for a large class of systems is given.

Analyticity and uniqueness for spin 1/2 classical ferromagnetic lattice systems at low temperatures

Analyticity and uniqueness of correlation functions is investigated for a number of systems by application of Ruelle's theorem on zeros of Asano contracted polynomials to the partition function. To answer the question when the partition function of a system is the Asano contraction of those of subsystems the groups appearing in the low and high temperature expansions are employed.

Multiparticle Generating Function and the Froissart Bound

Rigorous bounds for ${\ensuremath{\sigma}}_{\mathrm{tot}}(s)$ and ${\ensuremath{\sigma}}_{\mathrm{el}}(s)$ in terms of the smallest zero of the multiparticle generating function are derived. It is then shown that, if the analogy between this generating function and the grand partition function of a gas can be carried through to the point of establishing analyticity properties analogous to those established by Ruelle for a classical gas, then the Froissart bound can be improved essentially by $\mathrm{ln}s$. It is pointed out that the Ruelle theorem is valid in the multiperipheral model, and the steps needed for studying it in the general case are briefly outlined.