The Dynamics of The Potts-Bethe Mapping over $\mathbb{Q}_p:$ The Case $p\equiv 2 \ (mod \ 3) $

TL;DR

This paper investigates the behavior of the Potts-Bethe mapping in a specific p-adic setting, revealing its regularity for primes p congruent to 2 modulo 3, enhancing understanding of p-adic dynamical systems.

Contribution

It establishes the regularity of the Potts-Bethe mapping over the p-adic numbers for primes p where p ≡ 2 (mod 3), a case not previously analyzed.

Findings

Proves regularity of the mapping for p ≡ 2 (mod 3)

Extends understanding of p-adic Potts models

Provides mathematical foundation for further p-adic dynamics studies

Abstract

In this paper, we study the dynamics of the Potts-Bethe mapping associated with the $p-$adic $q-$state Potts model over the Cayley tree of order three. Namely, we establish the regularity of the Potts-Bethe mapping for the case $p\equiv 2 \ (mod \ 3)$ with $p\geq5$.