Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature

摘要

We generalize several results of the classical theory of ThermodynamicFormalism by considering a compact metric space $M$ as the state space. Weanalyze the shift acting on $M^\mathbb{N}$ and consider a general a-prioriprobability for defining the Transfer (Ruelle) operator. We study potentials$A$ which can depend on the infinite set of coordinates in $M^\mathbb{N}.$ We define entropy and by its very nature it is always a nonpositive number.The concepts of entropy and transfer operator are linked. If M is not a finiteset there exist Gibbs states with arbitrary negative value of entropy.Invariant probabilities with support in a fixed point will have entropy equalto minus infinity. In the case $M=S^1$, and the a-priori measure is Lebesgue$dx$, the infinite product of $dx$ on $(S^1)^\mathbb{N}$ will have zeroentropy. We analyze the Pressure problem for a H\"older potential $A$ and its relationwith eigenfunctions and eigenprobabilities of the Ruelle operator. Among otherthings we analyze the case where temperature goes to zero and we show someselection results. Our general setting can be adapted in order to analyze theThermodynamic Formalism for the Bernoulli space with countable infinitesymbols. Moreover, the so called XY model also fits under our setting. In thislast case M is the unitary circle $S^1$. We explore the differentiablestructure of $(S^1)^\mathbb{N}$ by considering potentials which are of class$C^1$ and we show some properties of the corresponding main eigenfunctions.