Translation-invariant Gibbs measures for the Blum–Kapel model on a Cayley tree

In the paper, translation-invariant Gibbs measures for the Blum–Kapel model on a Cayley tree of order $k$ are considered. An approximate critical temperature $T_{cr}$ is found such that for $T\geq T_{cr}$ there exists a unique translation-invariant Gibbs measure and for $0<T<T_{cr}$ there are exactly three translation-invariant Gibbs measures. In addition, the problem of (not) extremality for the unique Gibbs measure is studied.