Phase Transitions for Suspension Flows

Abstract

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.

Phase Transitions of the Multifractal Spectrum

“…Then, they prove that the map t → P(tg) has 0 to 1 phase transition when the roof function dominates the floor function. Using these results, Iommi and Jordan prove that the multifractal spectrum has 0 to 2 phase transitions in their paper [IJ2].…”

“…Their result is that the multifractal spectrum has 0 to 2 phase transitions. In the paper [IJ2], they take g to be a continuous function defined on the range of the suspension flow. Then, they prove that the map t → P(tg) has 0 to 1 phase transition when the roof function dominates the floor function.…”

“…Their paper [IJ1], considers level sets of [0, 1] generated by Birkhoff averages and expanding interval maps that have countably many branches. This paper uses results from [IJ2] and [IJ3]. Their results include a variational characterisation of the multifractal spectrum and the existence of 0 to 2 phase transitions for f µ (α) when α lim , which is a ratio involving the potentials φ and ψ, equals 0.…”

Measures of maximal entropy for suspension flows over the full shift

“…Examples of suspension flows with phase transitions at the zero potential (i.e. examples for which there does not exist a unique MME) have previously been obtained when the alphabet is infinite by Iommi, Jordan and Todd [10,11], and when the roof function is allowed to have zeroes by Savchenko [19]. In these examples, the phase transition occurs because of non-existence of an MME rather than non-uniqueness.…”

Princípio dos grandes desvios para estados de Gibbs-equilíbrio sobre shifts enumeráveis à temperatura zero

“…De fato, a prova apresentada utiliza o fato da medida de equilíbrio ser Gibbs, o que nos obriga a assumir que o shift tem a propriedade BIP (ver [Sar03]). É consenso entre especialistas em Formalismo Termodinâmico que o comportamento de shifts BIP é similar ao de shifts compactos [IJ13]. Nosso trabalho é mais um resultado que conrma esta expectativa.…”