Gibbs measures for the HC Blume–Capel model with countably many states on a Cayley tree

We study the Blume–Capel model with a countable set of spin values and a force of interaction between the nearest neighbors on a Cayley tree of order . The following results are obtained. Let , , be the temperature. For , there exist no translation invariant Gibbs measures or -periodic Gibbs measures. For , we prove the uniqueness of a translation-invariant Gibbs measure. Let and . If , then there exists exactly one -periodic Gibbs measure that is translation invariant. For , there exist exactly three -periodic Gibbs measures, one of which is a translation-invariant Gibbs measure.