Spectral properties of the Ruelle operator for product-type potentials on shift spaces

Cioletti, L.; Denker, Manfred; Lopes, Artur O.; Stadlbauer, Manuel

doi:10.1112/jlms.12031

Cited by 19 publications

(27 citation statements)

References 18 publications

(33 reference statements)

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“…In this case our numerical data suggest that z n (x), for large n, will be closed to an eigenfunction which is not the function g (see Figure 7). In [CDLS17] it is proved for this cases the existence of more than one measurable eigenfunctions. Figure 9: The graphs of the eigenfunction ϕ and of the approximation obtained via the involution kernel when γ = 3.3.…”

Section: Numerical Datamentioning

confidence: 96%

“…., where k n≥k |a n | < ∞. A large class of such potentials were carefully studied in [CDLS17] and spectral properties of the Ruelle operator were obtained there. We claim that A * = A (for some choice of W ).…”

Section: Involution Kernel Representations Of Eigenfunctionsmentioning

confidence: 99%

“…(2) See [CDLS17] for the computation of the topological pressure of B. In some sense we can think of this model as a simplified version of the previous one.…”

Section: Introductionmentioning

confidence: 99%

“…on the Section 8 provides a numerical comparison between the approximations computed by using the involution kernel and the explicit expression of the eigenfunction for some product type potentials, see[CDLS17].…”

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confidence: 99%

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Correlation inequalities and monotonicity properties of the Ruelle operator

Cioletti

Lopes

2019

Stoch. Dyn.

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Let X = {1, −1} N be the symbolic space endowed with a partial order , where x y, if x j ≥ y j , for all j ∈ N. A function f : X → R is called increasing if any pair x, y ∈ X, such that x y, we have f (x) ≥ f (y). A Borel probability measure µ over X is said to satisfy the FKG inequality if for any pair of continuous increasing functions f and g we have µ(f g) − µ(f )µ(g) ≥ 0. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice N and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions.As is widely known, for some continuous potentials does not exists a positive continuous eigenfunction associated to the spectral radius of the Ruelle operator acting on C(X). Here we employed some ideas related to the involution kernel in order to solve the main eigenvalue problem in a suitable sense for a class of potentials having low regularity. From this we obtain an explicit tight upper bound for the main eigenvalue (consequently for the pressure) of the Ruelle operator associated to Ising models with 1/r 2+ε interaction energy. Extensions of the Ruelle operator to suitable Hilbert Spaces are considered and a theorem solving to the main eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram theorem. We generalize results due to P. Hulse on attractive g-measures.We also present the graph of the main eigenfunction in some examples -in some cases the numerical approximation shows the evidence of not being continuous.

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Section: Numerical Datamentioning

confidence: 96%

Section: Involution Kernel Representations Of Eigenfunctionsmentioning

confidence: 99%

“…(2) See [CDLS17] for the computation of the topological pressure of B. In some sense we can think of this model as a simplified version of the previous one.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Correlation inequalities and monotonicity properties of the Ruelle operator

Cioletti

Lopes

2019

Stoch. Dyn.

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“…Continuous potentials not having continuous eigenfunctions. Following the results of [9] now we assume that the state space M = {−1, 1} and the a priori measure is the uniform measure, which we denote by κ. Let ρ be the infinite product measure ρ = i∈N κ.…”

Section: Using the Previous Three Inequalities We Get Formentioning

confidence: 99%

Ruelle operator for continuous potentials and DLR-Gibbs measures

Cioletti¹,

Lopes²,

Stadlbauer³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability.Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.

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Symmetric Measures, Continuous Networks, and Dynamics

Bezuglyi

Jørgensen

2021

New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative

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Spectral properties of the Ruelle operator for product-type potentials on shift spaces

Cited by 19 publications

References 18 publications

Correlation inequalities and monotonicity properties of the Ruelle operator

Correlation inequalities and monotonicity properties of the Ruelle operator

Ruelle operator for continuous potentials and DLR-Gibbs measures

Symmetric Measures, Continuous Networks, and Dynamics

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