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arXiv · 1909.04284

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

Farrukh Mukhamedov·Otabek Khakimov
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Abstract

In our previous investigations, we have developed the renormalization group method to $p$-adic $q$-state Potts model 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 parameters $q,k$. In \cite{MFKh18} 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. The present paper is a continuation of the mentioned 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 Markov partition. Moreover, the existence of Julia set is established, over which the mapping enables a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).

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Farrukh Mukhamedov, Otabek Khakimov. 2019-09-10. Chaotic behavior of the $p$-adic Potts-Bethe mapping II. https://arxiv.org/abs/1909.04284

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