Zero Temperature Limits of Gibbs Equilibrium States for Countable Markov Shifts

Kempton, Tom

doi:10.1007/s10955-011-0195-x

Cited by 26 publications

(44 citation statements)

References 13 publications

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“…This question has been positively answered by Contreras [9] in the case of expanding maps and Lipschitz continuous potentials by building up on previous results of Moris [23]. Finally we note that ground states and zero-temperature measures have also been studied in the context of countable Markov shifts by Kempton [13] and Jenkinson, Mauldin and Urbanski [17].…”

mentioning

confidence: 59%

Ground states and zero-temperature measures at the boundary of rotation sets

Kucherenko¹,

Wolf²

2017

Ergod. Th. Dynam. Sys.

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We consider a continuous dynamical system f : X → X on a compact metric space X equipped with an m-dimensional continuous potential Φ = (φ1, · · · , φm) : X → R m . We study the set of ground states GS(α) of the potential α · Φ as a function of the direction vector α ∈ S m−1 . We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of Φ. In particular, for each α the set of rotation vectors of GS(α) forms a non-empty, compact and connected subset of a face Fα(Φ) of the rotation set associated with α. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in Fα(Φ). We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any m ∈ N examples with an exposed boundary point (i.e. Fα(Φ) being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face Fα(Φ) with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of GS(α) is a non-trivial line segment.

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confidence: 59%

Ground states and zero-temperature measures at the boundary of rotation sets

Kucherenko¹,

Wolf²

2017

Ergod. Th. Dynam. Sys.

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“…The next proposition shows the existence of ground states. Some interesting results in this direction can be found in [3,5,9,10,12,13]. Therefore, follows of compactness of R N , that (µ βA ′ ) β>0 has an accumulation point µ ′ ∞ at infinity, in other words, there is a sequence (β n ) n∈N with β n → ∞ such that lim n∈N µ βnA ′ = µ ′ ∞ .…”

Section: C) Choosing Adequately the Eigenfunction ψ A We Have That ρmentioning

confidence: 93%

“…In some cases it is required to study these type of problems when the lattice is such that in each site the set of possible spins is unbounded. When the set of spins is countable several results are already known (see [9], [4], [10], [11], [12], [8] and section 5 in [14] )…”

Section: Introductionmentioning

confidence: 99%

Gibbs states and Gibbsian specifications on the space ℝ^ℕ

Lopes

Vargas

2019

Dynamical Systems

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We are interested in the study of Gibbs and equilbrium probabilities on the lattice R N . Consider the unilateral full-shift defined on the noncompact set R N and an α-Hölder continuous potential A from R N into R. From a suitable class of a priori probability measures ν (over the Borelian sets of R) we define the Ruelle operator associated to A (using an adequate extension of this operator to the compact set R N = (S 1 ) N ) and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to A. The above, can be seen as a generalization of the results obtained in the compact case for the XY-model. We also introduce an extension of the definition of entropy and show the existence of A-maximizing measures. Moreover, we prove the existence of an involution kernel for A. Finally, we build a Gibbsian specification for the Borelian sets on the set R N and we show that this family of probability measures satisfies a FKGinequality.

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“…For the case of (X, T ) a countable alphabet subshift of finite type, where X is non-compact and the entropy map µ → h(µ) is not upper semi-continuous, additional summability and boundedness hypotheses on the locally Hölder function f : X → R, together with primitivity assumptions on X, ensure existence and uniqueness of the equilibrium measures m tf , that the family (m tf ) does in fact have an accumulation point m, and that h(m) = lim t→∞ h(m tf ) = max{h(µ) : µ ∈ M max (f )} (see [62,92,123]), representing an analogue of Theorem 4.1. If in addition f is locally constant, Kempton [101] (see also [62]) has established the analogue of Theorem 4.2, guaranteeing the weak * convergence of (m tf ) as t → ∞. Iommi & Yayama [79] consider almost additive sequences F of continuous functions defined on appropriate countable alphabet subshifts of finite type, proving that the family of equilibrium measures (m tF ) is tight (based on [92]), hence has a weak * accumulation point, and that any such accumulation point is a maximizing measure for F (see also [47,63,150,161] for general ergodic optimization in the context of sequences of functions F).…”

Section: Ergodic Optimization As Zero Temperature Thermodynamic Formamentioning

confidence: 99%

Ergodic optimization in dynamical systems

Jenkinson¹

2018

Ergod. Th. Dynam. Sys.

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Abstract. Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called f -maximizing if the time average of the real-valued function f along the orbit is larger than along all other orbits, and an invariant probability measure is called fmaximizing if it gives f a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

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Zero Temperature Limits of Gibbs Equilibrium States for Countable Markov Shifts

Cited by 26 publications

References 13 publications

Ground states and zero-temperature measures at the boundary of rotation sets

Ground states and zero-temperature measures at the boundary of rotation sets

Gibbs states and Gibbsian specifications on the space ℝ^ℕ

Ergodic optimization in dynamical systems

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Zero Temperature Limits of Gibbs Equilibrium States for Countable Markov Shifts

Cited by 26 publications

References 13 publications

Ground states and zero-temperature measures at the boundary of rotation sets

Ground states and zero-temperature measures at the boundary of rotation sets

Gibbs states and Gibbsian specifications on the space ℝℕ

Ergodic optimization in dynamical systems

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Gibbs states and Gibbsian specifications on the space ℝ^ℕ