Phase transition and chaos: P-adic Potts model on a Cayley tree

Abstract

In our previous investigations, we have developed the renormalization group method to p-adic models on Cayley trees, this method is closely related to the investigation of dynamical system associated with a given model. In this paper, we are interested in the following question: how is the existence of the phase transition related to chaotic behavior of the associated dynamical system (this is one of the important question in physics)? To realize this question, we consider as a toy model the p-adic q-state Potts model on a Cayley tree, and show, in the phase transition regime, the associated dynamical system is chaotic, i.e. it is conjugate to the full shift. As an application of this result, we are able to show the existence of periodic (with any period) p-adic quasi Gibbs measures for the model. This allows us to know that how large is the class of p-adic quasi Gibbs measures. We point out that a similar kind of result is not known in the case of real numbers.