On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection

We consider $(M,d)$ a connected and compact manifold and we denote by ${\mathcal{B}}_i$ the Bernoulli space $M^{\mathbb{Z}}$ of sequences represented by $$x=(\mathrm{.}\mathrm{.}\mathrm{.}{x}_{-3},x_{-2},x_{-1},x_0,x_1,x_2,x_3,\mathrm{.}\mathrm{.}\mathrm{.}),$$ where $x_i$ belongs to the space (alphabet) $M$. The case where $M={\mathbb{S}}^1$, the unit circle, is of particular interest here. The analogous problem in the one-dimensional lattice $\mathbb{N}$ is also considered. %In this case we consider the potential $A:\mathcal{B}=M^{\mathbb{N}}\to \mathbb{R}\mathrm{.}$ Let $A: $\mathcal{B}$_i \rar \R$beanobservableorpotentialdefinedintheBernoullispace$$\mathcal{B}$_i$\mathrm{.}Thepotential$A$ describes an interaction between sites in the one-dimensional lattice $M^{\mathbb{Z}}$. Given a temperature $T$, we analyze the main properties of the Gibbs state ${\widehat{\mu }}_{\frac{1}{T}A}$ which is a certain probability measure over ${\mathcal{B}}_i$. We denote this setting “the general XY model”. In order to do our analysis we consider the Ruelle operator associated to $\frac{1}{T}A$, and, we get in this procedure the main eigenfunction ${\psi }_{\frac{1}{T}A}$. Later, we analyze selection problems when temperature goes to zero: a) existence, or not, of the limit (on the uniform convergence) $$V:=\underset{T\to 0}{\lim }T\log ({\psi }_{\frac{1}{T}A}),\text{a question about selection of subaction},$$ and, b) existence, or not, of the limit (on the weak${}^{\ast }$ sense) $$\tilde{\mu }:=\underset{T\to 0}{\lim }{\widehat{\mu }}_{\frac{1}{T}A},\text{a question about selection of measure}\mathrm{.}$$ The existence of subactions and other properties of Ergodic Optimization are also considered.