A HC model with countable set of spin values: Uncountable set of Gibbs measures

We consider a hard core (HC) model with a countable set Z of spin values on the Cayley tree. This model is defined by a countable set of parameters lambda(i) > 0, i is an element of Z\{0}. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.