On chaos of a cubic $p$-adic dynamical system

On the chaotic behavior of a generalized logistic $p$-adic dynamical system

In the paper we describe basin of attraction $p$-adic dynamical system $G(x)=(ax)^2(x+1)$. Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the $p$-adic Siegel discs.

关于三次2-adic Mandelbrot集的一个切片

On a Slice of the Cubic 2-adic Mandelbrot Set

本文研究三次2-adic Mandelbrot集的一个单参数切片，证明t=1处为非孤立边界点并具有渐近自相似性，同时揭示该点处Mandelbrot集与Julia集之间的相似性。

Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures

We study the set of $p$-adic Gibbs measures of the $q$-states Potts model on the Cayley tree of order three. We prove the vastness of the periodic $p$-adic Gibbs measures for such model by showing the chaotic behavior of the correspondence Potts--Bethe mapping over $\mathbb{Q}\_p$ for $p\equiv 1 \ (\rm{mod} \ 3)$. In fact, for $0 < |θ-1|\_p < |q|\_p^2 < 1$, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for $0 < |q|\_p^2 \leq |θ-1|\_p < |q|\_p < 1$, there exists a subsystem that isometrically conjugate to a subshift of finite type on $r$ sy…

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

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 a…

Quadratic non-stochastic operators: Examples of splitted chaos

There is one-to-one correspondence between quadratic operators (mapping $\mathbb R^m$ to itself) and cubic matrices. It is known that any quadratic operator corresponding to a stochastic (in a fixed sense) cubic matrix preserves the standard simplex. In this paper we find conditions on the (non-stochastic) cubic matrix ensuring that corresponding quadratic operator preserves simplex. Moreover, we construct several quadratic non-stochastic operators which generate chaotic dynamical systems on the simplex. These chaotic behaviors are \emph{splitted} meaning that the simplex is partitioned into u…

与 $\n\b\mathbb{Q}_p$ 上射影直线上的 Sigmoid Beverton-Holt 模型相关的有理映射

Rational map associated with the Sigmoid Beverton-Holt model on the projective line over $\mathbb{Q}_p$

本文利用 $p$-进多项式最小分解及 $p$-进排斥子混沌理论，刻画了 $\n\b\mathbb{Q}_p$ 射影直线上 Sigmoid Beverton-Holt 模型对应有理映射的动力学结构。