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 of the attractiveness of the cycle.
(ii) In the case where the fixed point $x_0$ is unique we prove that if the point is attracting then there exists $δ>0$, such that the basin of attraction for $x_0$ is the ball of radius $δ$ and the center at $x_0$ and any sphere with radius $\geq δ$ is invariant.
If $x_0$ is an indifferent point then all spheres with the center at $x_0$ are invariant. If $x_0$ is a repelling point then there exits $δ>0$, such that the trajectory which starts at an element of the ball of radius $δ$ with the center in $x_0$ leaves this ball, whereas any sphere with radius $\geq δ$ is invariant.
(iii) In case of existence of two fixed points, we show that Siegel disks may either coincide or be disjoint for different fixed points of the dynamical system. Besides, we find the basin of the attractor of the system. Varying the parameters it is proven that there exists an integer $k\geq 2$, and spheres $S_{r_1}(x_i), ..., S_{r_k}(x_i)$ such that the limiting trajectory will be periodically traveling the spheres $S_{r_j}$. For some values of the parameters there are trajectories which go arbitrary far from the fixed points.
We consider a family of $(2,1)$-rational functions given on the set of $p$-adic field $Q_p$. Each such function has a unique fixed point. We study ergodicity properties of the dynamical systems generated by $(2,1)$-rational functions. For each such function we describe all possible invariant spheres. We characterize ergodicity of each $p$-adic dynamical system with respect to Haar measure reduced on each invariant sphere. In particular, we found an invariant spheres on which the dynamical system is ergodic and on all other invariant spheres the dynamical systems are not ergodic.
We consider $(1,2)$-rational functions given on the field of $p$-adic numbers $\mathbb Q_p$. In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then $(1,2)$-rational function is conjugate to a two-parametric $(1,2)$-rational function. Depending on these two parameters we determine the type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study the ergodicity properties of the function on each invariant set. We de…
In this paper we consider dynamical systems generated by $(3,2)$-rational functions on the field of $p$-adic complex numbers. Each such function has three fixed points. We show that Siegel disks of the dynamical system may either coincide or be disjoint for different fixed points. Also, we find the basin of each attractor of the dynamical system. We show that, for some values of the parameters, there are trajectories which go arbitrary far from the fixed points.
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)…
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…
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…
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…
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…
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 $a$.
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…
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…
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 $p\geq 3$.
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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…
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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 …
