Entropy and variational principle for one-dimensional lattice systems with a generala prioriprobability: positive and zero temperature

• DOI: 10.1017/etds.2014.15
• Publication Date: 2014-07-03T10:01:27Z
• Authors: A. O. LOPES; J. K. MENGUE; J. MOHR; R. R. SOUZA
• Keywords: Eigenfunction

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