Periodic Gibbs Measures and Their Extremality for the HC-Blume–Capel Model in the Case of a Wand with a Chemical Potential on a Cayley Tree

Periodic Gibbs measures for the HC-Blume–Capel model with a chemical potential with parameters $(\theta,\eta)$ on a Cayley tree in the case of a wand graph are studied. We prove that in this case for $\theta^3\leqslant\eta$ there exist precisely three periodic Gibbs measures, all of which are translation-invariant, while for $\theta^3>\eta$ there exist precisely three periodic Gibbs measures, one of which is translation-invariant and the other two are $2$-periodic (but not translation-invariant). The (non)extremality of these measures is also studied.