Fixed points of an infinite-dimensional operator related to Gibbs measures.

We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set of spin values on a Cayley tree. This operator is defined by a countable set of parameters , , . We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model. [ABSTRACT FROM AUTHOR]