Classification of Fixed Points of Potts–Bethe Mapping of Degree Four on Q5

Abstract

The Potts–Bethe mapping is a rational function arises in the study of the Potts model on the Cayley tree (or Bethe lattice). In this paper, the Potts–Bethe mapping of degree four is considered over the field Q₅ of 5-adic numbers. In some regimes (a condition appear in the study of p-adic Potts model), the fixed points are found and their stability are determined. It is done by solving some quartic equation over Q₅ and calculating the value of derivative at each fixed points. This is the continuation of the previous work where contraction and chaos are found, but here other property is realized such as 1-Lipschitz.