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

Abstract

The 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).

Описание обобщенных трансляционно-инвариантных $p$-адических мер Гиббса для модели Поттса на дереве Кэли порядка три

“…Следует отметить, что динамическое поведение отображения (4.3) ранее уже изучалось в литературе (см. [18], [35], [36], [39], [40]). Рассмотрим случай k = 3 и предположим, что |A| = 1.…”

“…Замечание 4.1. Теорема 4.1 согласуется с описанием обобщенных трансляционно-инвариантных p-адических мер Гиббса, ранее полученным в [18], [35], [36], для случаев 0…”

“…Следует отметить (см. замечание 4.1), что основной результат нашей статьи подтверждает и расширяет родственные результаты в литературе [18], [35], [36].…”

Stochastic-like characteristics of arithmetic dynamical systems: the Collatz hailstone sequences

“…By its turn, p-adic numbers (simply put, the decomposition of an integer as a power series of a prime p, see next sections) are very useful technical tools in different areas of physics as quantum mechanics [15,16], quantum logic [17], gravity [18], and string theory [19]. Further, they have been employed in the study of stochastic and complex systems, like in self-organized criticality (SOC) [20], phase transitions in Ising [21,22] and Potts [23][24][25][26] models, as well as in Gibbs measures [21,27,28], Markov processes [29], and diffusion in random media [30]. For a complete review on p-adics usages see, e.g.…”

p$$ p $$‐Adic phase transitions for countable state Potts model

“…First, $$$ p $$$‐adic lattice models have been appeared in the literature [24–28] to investigate phase transition issues of certain $$$ p $$$‐adic models over Cayley trees (see Mukhamedov and Khakimov [23] for recent review). In previous works [29–36], using chaotic behavior of the $$$ p $$$‐adic renormalization group (dynamical systems approach) technique, it has been established vastness of the set of $$$ p $$$‐adic Gibbs measures for the Potts models on a Cayley tree. Recently, in Rozikov [37], a comprehensive treatment with applications of the Potts model has been presented.…”