Ground states are generically a periodic orbit

Contreras, Gonzalo

doi:10.1007/s00222-015-0638-0

Cited by 59 publications

(68 citation statements)

References 22 publications

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“…6 In dimension d = 1, Brémont showed that there is no finite-range interaction that is chaotic, see [7] and also [8,16,22]. See also [3,5,11,17] and the monograph [4] for recent related results.…”

Section: Symbolic Spacementioning

confidence: 96%

Sensitive Dependence of Gibbs Measures at Low Temperatures

Coronel

Rivera-Letelier

2015

J Stat Phys

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The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

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Section: Symbolic Spacementioning

confidence: 96%

Sensitive Dependence of Gibbs Measures at Low Temperatures

Coronel

Rivera-Letelier

2015

J Stat Phys

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“…C 1 ) functions on X, though the case V = Lip was discussed in more detail. In subsequent years this case V = Lip became a focus of attention among workers in ergodic optimization, culminating in its resolution (see Theorem 7.10 below) by Contreras [50], building on work of [125,141,159].…”

Section: Typically Periodic Optimization (Tpo)mentioning

confidence: 99%

“…16 Condition (16) was introduced by Yuan & Hunt [159, p. 1217], who called it the Class I condition. (A Class II condition was also introduced in [159], in terms of approximability by periodic orbit measures, which stimulated related work in [42,49] [50] showed, by estimating the lengths of pseudo-orbits, that if µ ∈ M T \ M Y H then µ has strictly positive entropy. However, a result of Morris [125] asserts that the set of Lipschitz functions with a positive entropy maximizing measure is of first category; it follows that Lip \ Lip Y H is of first category, and therefore Lip Y H is dense in Lip, as required.…”

Section: Typically Periodic Optimization (Tpo)mentioning

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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“…Some examples of this are [9,7,6,5,3,10,11]. A good introduction to the subject is [8], where the fundamental results of the theory are displayed and recentelly, in [4] an important conjecture was proved.…”

Section: Introductionmentioning

confidence: 99%