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Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach.
"... We consider a family of potentials f, derived from the Hofbauer poten-tials, on the symbolic space Ω = {0, 1}N and the shift mapping σ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pres-sure is not a ..."
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We consider a family of potentials f, derived from the Hofbauer poten-tials, on the symbolic space Ω = {0, 1}N and the shift mapping σ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pres-sure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the Thermo-dynamic Limit is proved by considering a version of a Renewal Equation. We also show that the correlations decay polynomially and compute the
Duality between Eigenfunctions and Eigendistributions of Ruelle and Koopman operators via an integral kernel, preprint Arxiv
, 2014
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Thieullen A thermodynamic formalism for continuous time Markov chains with values on the Bernoulli Space: entropy, pressure and large deviations,
- Journ. of Statist. Phys.
, 2013
"... ABSTRACT. Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice {1, . . . , d} N (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator L = L A − I, where L A is a discre ..."
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ABSTRACT. Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice {1, . . . , d} N (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator L = L A − I, where L A is a discrete time Ruelle operator (transfer operator), and A : {1, . . . , d} N → R is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the a priori probability. Given a Lipschitz interaction V : {1, . . . , d} N → R, we are interested in Gibbs (equilibrium) state for such V . This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V . Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative (see also
Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces
"... Abstract Recently the Ruelle-Perron-Fröbenius theorem was proved for Hölder potentials defined on the symbolic space Ω = M N , where (the alphabet) M is any compact metric space. In this paper, we extend this theorem to the Walters space W (Ω), in similar general alphabets. We also describe in deta ..."
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Abstract Recently the Ruelle-Perron-Fröbenius theorem was proved for Hölder potentials defined on the symbolic space Ω = M N , where (the alphabet) M is any compact metric space. In this paper, we extend this theorem to the Walters space W (Ω), in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fréchet-analyticity of the Ruelle operator under quite general conditions and we apply this result to prove the analytic dependence of this operator on both Walters and Hölder spaces. The analyticity of the pressure functional on Hölder spaces is established. An exponential decay of the correlations is shown when the Ruelle operator has the spectral gap property. A new (and natural) family of Walters potentials (on a finite alphabet derived from the Ising model) not having an exponential decay of the correlations is presented. Because of the lack of exponential decay, for such potentials we have the absence of the spectral gap for the Ruelle operator. The key idea to prove the lack of exponential decay of the correlations are the Griffiths-Kelly-Sherman inequalities.
