On the Existence of Generalized Gibbs Measures for the One-Dimensional $p$-adic Countable State Potts Model

We consider the one-dimensional countable state $p$-adic Potts model. A construction of generalized $p$-adic Gibbs measures depending on weights $\lambda$ is given, and an investigation of such measures is reduced to the examination of a $p$-adic dynamical system. This dynamical system has a form of series of rational functions. Studying such a dynamical system, under some condition concerning weights, we prove the existence of generalized $p$-adic Gibbs measures. Note that the condition found does not depend on the values of the prime $p$, and therefore an analogous fact is not true when the number of states is finite. It is also shown that under the condition there may occur a phase transition.