On a Class of Rational $P$-Adic Dynamical Systems

具有空Fatou集的p进有理映射

$p$-adic rational maps having empty Fatou set

本文证明了在p进数域任意有限代数扩张上，存在有理映射使得其动力系统的Fatou集为空，即迭代族处处不等度连续。

定位吸引 p 进吸引周期的关键点

Locating critical points attracted to p-adic attracting cycles

本文研究非阿基米德动力学中吸引周期吸引临界点的条件，通过分析乘数建立精确判定方法。

$p$-Adic interpolation of orbits under rational maps

Let $L$ be a field of characteristic zero, let $h:\mathbb{P}^1\to \mathbb{P}^1$ be a rational map defined over $L$, and let $c\in \mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and $h$ are defined along with an infinite set of inequivalent non-archimedean completions $K_{\mathfrak{p}}$ for which there exists a positive integer $a=a(\mathfrak{p})$ with the property that for $i\in \{0,\ldots ,a-1\}$ there exists a power series $g_i(t)\in K_{\mathfrak{p}}[[t]]$ that converges on the closed unit disc of $K_{\mathfrak{p}}$ such that $h^{an+i…

Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the…

On one polynomial $p$-adic dynamical system

In the paper we describe basin of attraction and the Siegel discs of the $p$-adic dynamical system $f(x)=x^{2n+1}+ax^{n+1}$ over complex $p$-adic field.

An adic dynamical system related to the Delannoy numbers

We introduce an adic (Bratteli-Vershik) dynamical system based on a diagram whose path counts from the root are the Delannoy numbers. We identify the ergodic invariant measures, prove total ergodicity for each of them, and initiate the study of the dimension group and other dynamical properties.

Discreteness of postcritically finite maps in p-adic moduli space

Let $p \geq 2$ be a prime number and let $\mathbb{C}_p$ be the completion of an algebraic closure of the $p$-adic rational field $\mathbb{Q}_p$. Let $f_c(z)$ be a one-parameter family of rational functions of degree $d\geq 2$, where the coefficients are meromorphic functions defined at all parameters $c$ in some open disk $D\subseteq\mathbb{C}_p$. Assuming an appropriate stability condition, we prove that the parameters $c$ for which $f_c$ is postcritically finite (PCF) are isolated from one another in the $p$-adic disk $D$, except in certain trivial cases. In particular, all PCF parameters of…

p-Adic and Adelic Rational Dynamical Systems

In the framework of adelic approach we consider real and p-adic properties of dynamical system given by linear fractional map f (x) = (a x + b)/(c x + d), where a, b, c and d are rational numbers. In particular, we investigate behavior of this adelic dynamical system when fixed points are rational. It is shown that any of rational fixed points is p-adic indifferent for all but a finite set of primes. Only for finite number of p-adic cases a rational fixed point may be attractive or repelling. The present analysis is a continuation of the paper math-ph/0612058. Some possible generalizations are…

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.

Rational maps are $d$-adic Bernoulli

Freire, Lopes and Mane proved that for any rational map f there exists a natural invariant measure μ_f [5]. Mane showed there exists an n>0 such that (f^n, μ_f) is measurably conjugate to the one-sided $d^n$-shift, with Bernoulli measure $(\frac 1{d^n},... ,\frac 1{d^n})$ \[15]. In this paper we show that (f,μ_f)is conjugate to the one-sided Bernoulli $d$-shift. This verifies a conjecture of Freire, Lopes and Mane [5] and Lyubich [11].

On minimal decomposition of $p$-adic polynomial dynamical systems

A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on $\mathbb{Z}_2$, we exhibit all its minimal subsystems.

$p$-adic dynamical systems of $(2,2)$-rational functions with unique fixed point

We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathbb{C}_p$. Each such function has a unique fixed point. We study $p$-adic dynamical systems generated by the $(2,2)$-rational functions. We show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. Siegel disks of these dynamical systems are found. We obtain an upper bound for the set of limit points of each trajectory, i.e., we determine a sufficiently small set containing the set of limit points. For each $(2,2)…

p-进次双曲有理映射的扩张性质与统计律

Expanding property and statistical laws for $p$-adic subhyperbolic rational maps

本文研究p-进数域上有理映射的次双曲性，证明其在奇异度量下具有扩张性质，并在温和假设下对具有紧C_p-Julia集的次双曲映射建立了若干统计律。

$p$-adic dynamical systems of $(3,1)$-rational functions with unique fixed point

We describe the set of all $(3,1)$-rational functions given on the set of complex $p$-adic field $\mathbb C_p$ and having a unique fixed point. We study $p$-adic dynamical systems generated by such $(3,1)$-rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each $(3,1)$-rational function on $\mathbb C_p$ there is a point $\hat x=\hat x(f)\in \mathbb…

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

In the paper we describe basin of attraction of the $p$-adic dynamical system $f(x)=x^3+ax^2$. 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.

Minimality of p-adic rational maps with good reduction

A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}\_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimal…

Preperiodic points with local rationality conditions in the quadratic unicritical family

For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic polynomials $f_c$ with rational parameters $c\in\mathbb{Q}$ so that $f_c$ has only finitely many totally real (totally $p$-adic, respectively) preperiodic points. These results rely on an adelic Fekete-type theorem and dynamics of the filled Julia set of $f_c$. Moreover, using a numerical criterion introduced in [NP], we make explicit calculations of the set of totall…

A $p$-adic Approach To Piecewise Polynomial Dynamical Systems

Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to \mathbb{Z}_+$ defined by $$\text{Col}(n) = \begin{cases} 3n + 1, & n \text{ is odd;} \\ n/2, & \text{otherwise.}\end{cases}$$ Proving the non-existence of non-trivial Collatz cycles would reduce the Collatz conjecture to whether the Collatz map can diverge to infinity.

关于高度为一的p进动力系统的卢宾猜想

Lubin's conjecture for height-one $p$-adic dynamical systems

利用p进霍奇理论，证明了关于\(\mathbb{Q}_p\)有限扩域整数环\(\mathcal{O}\)上高度为一的交换形式幂级数对的卢宾猜想，表明这样的一对是\(\mathcal{O}\)上形式群的自同态对。

Criteria of measure-preserving for $p$-adic dynamical systems in terms of the van der Put basis

This paper is devoted to (discrete) $p$-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of $p$-adic dynamical systems. Given continuous function $f:Z_p > Z_p.$ Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary $p$? In this paper, for any prime $p$ we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterio…

On $p$-adic entropy of some solenoid dynamical systems

To a dynamical system is attached a non-negative real number called entropy. In 1990, Lind, Schmidt and Ward proved that the entropy for the dynamical system induced by the Laurent polynomial algebra over the ring of the rational integers is described by the Mahler measure. In 2009, Deninger introduced the $p$-adic entropy and obtained a $p$-adic analogue of Lind-Schmidt-Ward's theorem by using the $p$-adic Mahler measures. In this paper, we prove the existence and the explicit formula about $p$-adic entropies for two dynamical systems; one is induced by the Laurent polynomial algebra over the…

Minimality of $5$-adic polynomial dynamical systems

We characterize the dynamical systems consisting of the set of 5-adic integers and polynomial maps which consist of only one minimal component.

Criteria of ergodicity for $p$-adic dynamical systems in terms of coordinate functions

This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The coordinate representation can be useful, e.g., for applications to cryptography. Moreover, by using this representation we can consider non-smooth $p$-adic transformations. The basic technical toolsare van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theore…

Dynamical systems of the $p$-adic $(2,2)$-rational functions with two fixed points

We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathcal{C}_p$. Each such function $f$ has the two distinct fixed points $x_1=x_1(f)$, $x_2=x_2(f)$. We study $p$-adic dynamical systems generated by the $(2,2)$-rational functions. We prove that $x_1$ is always indifferent fixed point for $f$, i.e., $x_1$ is a center of some Siegel disk $SI(x_1)$. Depending on the parameters of the function $f$, the type of the fixed point $x_2$ may be any possibility: indifferent, attractor, repeller. We find Siegel disk or basin of attraction of the fixed point $x…

$p$-adic dynamical systems of the function $a x^{-2}$

In this paper we study $p$-adic dynamical systems generated by the function $f(x)={a\over x^2}$ in the set of complex $p$-adic numbers. We find an explicit formula for the $n$-fold composition of $f$ for any $n\geq 1$. Using this formula we give fixed points, periodic points, basin of attraction and Siegel disk of each fixed (periodic) point depending on parameters $p$ and $a$.

On ergodic behavior of $p$-adic dynamical systems

Monomial mappings, $x\mapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The process is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1.

Rational map ax+1/x on the projective line over $\mathbb{Q}_2$

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.

Cayley树上p-adic λ-模型的p-adic动力系统

p-adic dynamical systems for $p$-adic $λ$-models on the Cayley trees

本文利用关联动力系统与重整化技术研究Cayley树上p-adic λ-模型的相变现象，通过广义p-adic拟吉布斯测度建立递推方程，并验证了动力系统视角下的预测。

$p$-adic dynamical systems of the function $\dfrac{ax}{x^2+a}$

We show that any $(1,2)$-rational function with a unique fixed point is topologically conjugate to a $(2,2)$-rational function or to the function $f(x)={ax\over x^2+a}$. The case $(2,2)$ was studied in our previous paper, here we study the dynamical systems generated by the function $f$ on the set of complex $p$-adic field $\mathbb C_p$. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set…

Rational map $ax+1/x$ on the projective line over $\mathbb{Q}\_{p}$

The dynamical structure of the rational map $ax+1/x$ on the projective line $¶$ over the field $\mathbb{Q}\_p$ of $p$-adic numbers is described for $p\geq 3$.

On minimal decomposition of $p$-adic homographic dynamical systems

A homographic map in the field of $p$-adic numbers $\mathbb{Q}_p}$ is studied as a dynamical system on $\mathbb{P}^{1}(\mathbb{Q}_p)$, the projective line over $\mathbb{Q}_p$. If such a system admits one or two fixed points in $\mathbb{Q}_p$, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the min…

Beyond substitutive dynamical systems: S-adic expansions

An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.

Dynamical properties of the Pascal adic transformation

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.

$p$-adic $(2,1)$-rational dynamical systems

We investigate the trajectory of an arbitrary $(2,1)$-rational $p$-adic dynamical system in a complex $p$-adic field $\C_p$. (i) In the case where there is no fixed point we show that the $p$-adic dynamical system has a 2-periodic cycle $x_1, x_2$. If it is attracting then it attracts each trajectory which starts from an element of a ball of radius $r=|x_1-x_2|_p$ with the center at $x_1$ or at $x_2$. If the 2-periodic cycle is an indifferent, then in each step the balls transfer to each other. All the other spheres with radius $>r$ and the center at $x_1$ and $x_2$ are invariant independently…

有限生成 S-adic 系统的上边缘与特征值

Coboundaries and eigenvalues of finitary S-adic systems

本文研究了有限生成可识别 S-adic 动力系统的特征值与上边缘，通过引入直性与本质词概念定义上边缘，给出了复数成为连续或可测特征值的充要条件，并应用于等长替换系统证明了连续特征值均为有理数。

On a non-linear $p$-adic dynamical system

We investigate the behavior of trajectories of a $(3,2)$-rational $p$-adic dynamical system in the complex $p$-adic filed ${\mathbb C}_p$, when there exists a unique fixed point $x_0$. We study this $p$-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point $x_0$). We show that there exists a radius $r$ depending on parameters of the rational function such that: when $x_0$ is an attracting point then the trajectory of an inner point from the ball $U_r(x_0)$ goes to $x_0$ and each sphere with a radius $>r$ (with the center at $x_0$) is invariant; When $x…

Automata as $p$-adic Dynamical Systems

The automaton transformation of infinite words over alphabet $\mathbb F_p=\{0,1,\ldots,p-1\}$, where $p$ is a prime number, coincide with the continuous transformation (with respect to the $p$-adic metric) of a ring $\mathbb Z_p$ of $p$-adic integers. The objects of the study are non-Archimedean dynamical systems generated by automata mappings on the space $\mathbb Z_p$. Measure-preservation (with the respect to the Haar measure) and ergodicity of such dynamical systems plays an important role in cryptography (e.g. for pseudorandom generators and stream cyphers design). The possibility to use …

离散动力学的有理解释器：$p$-adic域上的存在性、精确性与分解

Rational Interpreters for Discrete Dynamics: Existence, Exactness, and Decomposition over $p$-adic Fields

本文研究了$p$-adic域上离散动力学的有理解释器的存在性、精确性与分解，提出动态中国剩余定理并探讨了逆极限扩展。

On dynamical systems and phase transitions for $Q+1$-state $P$-adic Potts model on the Cayley tree

In the present paper, we introduce a new kind of $p$-adic measures for $q+1$-state Potts model, called {\it $p$-adic quasi Gibbs measure}. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define one dimensional fractional $p$-adic dynamical system. In ferromagnetic case, we establish that if $q$ is divisible by $p$,…

The Euler adic dynamical system and path counts in the Euler graph

We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that the natural symmetric measure for the Bratteli-Vershik dynamical system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex.

Group structure of the $p$-adic ball and dynamical system of isometry on a sphere

In this paper the group structure of the $p$-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations $\oplus$ and $\odot$ on a ball and sphere respectively, and prove that this sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in $\mathbb Z_p$ is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometr…