Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree

摘要

We continue our study of the full set of translation-invariant splittingGibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) forthe $q$-state Potts model on a Cayley tree. In our previous work \cite{KRK} wegave a full description of the TISGMs, and showed in particular that atsufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuringthat a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which thereare at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numericalvalues.