For the q-state Potts model on a Cayley tree of order k ≥ 2 it is wellknown that at sufficiently low temperatures there are at least q +1 translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs).In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is 2 q − 1. We prove that there are [q/2] (where [a] is the integer part of a) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs.While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

We study translation-invariant splitting Gibbs measures (TISGMs, treeindexed Markov chains) for the fertile three-state hard-core models with activity λ > 0 on the Cayley tree of order k ≥ 1. There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearestneighbor exclusion. It is known that (i) for the wrench and pipe cases ∀λ > 0 and k ≥ 1, there exists a unique TISGM; (ii) for hinge (resp. wand) case at k = 2 if λ < λ cr = 9/4 (resp. λ < λ cr = 1), there exists a unique TISGM, and for λ > 9/4 (resp. λ > 1), there exist three TISGMs. In this paper we generalize the result (ii) for any k ≥ 2, i.e., for hinge and wand cases we find the exact critical value λ cr (k) with properties mentioned in (ii). Moreover, we find some regions for the λ parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.

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