Flexibility of the Pressure Function
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Publication:6374235
DOI10.1007/S00220-022-04466-YarXiv2108.00451MaRDI QIDQ6374235
Tamara Kucherenko, Anthony Quas
Publication date: 1 August 2021
Abstract: 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 emph{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.
Smooth ergodic theory, invariant measures for smooth dynamical systems (37C40) Thermodynamic formalism, variational principles, equilibrium states for dynamical systems (37D35) Symbolic dynamics (37B10)
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