Chaotic behavior of the p-adic Potts–Bethe mapping II

Otabek Khakimov,Farrukh Mukhamedov

doi:10.1017/etds.2021.96

Abstract

AbstractThe renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).

Highlights

The presentpaper is a continuation of [35], where we have started to investigate the chaotic behavior of the Potts–Bethe mapping over the p-adic field.Downloaded from https://www.cambridge.org/core. 06 Jan 2022 at 09:31:26, subject to the Cambridge Core terms of use.O
Under some conditions, we were able to prove that fθ,q,k is conjugate to the full shift on κp symbols (here κp is the greatest common factor (GCF) of k and p − 1)
It is known that the thermodynamic behavior of the central site of the Potts model with nearest-neighbor interactions on a Cayley tree is reduced to the recursive system which is given by (1.1)

Summary

Introduction

The presentpaper is a continuation of [35], where we have started to investigate the chaotic behavior of the Potts–Bethe mapping over the p-adic field (here p is some prime number).O. If κp is the GCF of k and p − 1, the following hold: (B1) if κp = 1, there exists x∗ ∈ Fix(fθ,q,k) such that x∗ = x0∗ and Jfθ,q,k = In this paper, we are able to prove the chaoticity of the Potts–Bethe mapping for arbitrary values of k (under the condition |θ − 1|p < |q|2p) and we are not even using the existence of the fixed points.

Objectives

Results

Conclusion