Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms

Carrasco, P.; Saghin, Radu

doi:10.48550/arxiv.2101.07089

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2021

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“…Subsequently, the fundamental paper [10] outlines the program and provides flexibility results for volume-preserving systems with respect to the volume measure. These results have already been improved and extended in [15].…”

Section: Overviewmentioning

confidence: 72%

Flexibility of the Pressure Function

Kucherenko¹,

Quas²

2021

Preprint

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We study the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. In fact, we establish a multidimensional version of this result. As a consequence, we obtain that for a continuous observable the phase transitions can occur at a countable dense set of temperature values. We go further and show that one can vary the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number, finite or infinite.

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Section: Overviewmentioning

confidence: 72%