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Phase transition and chaos: $$p$$-adic Potts model on a Cayley tree. (English) Zbl 1355.37106
Summary: 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.

##### MSC:
 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 39A70 Difference operators 47S10 Operator theory over fields other than $$\mathbb{R}$$, $$\mathbb{C}$$ or the quaternions; non-Archimedean operator theory 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics
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