arXiv subjects

I. A. Sattarov

Publications and source records attributed to I. A. Sattarov.

11 recordsLinked to original sources

arXiv2023

Ergodicity and periodic orbits of $p$-adic $(1,2)$-rational dynamical systems with two fixed points

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 describe 2- and 3-periodic orbits of the $p$-adic dynamical systems generated by the two-parametric $(1,2)$-rational functions.

math.DS↗

arXiv2022

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 isometry defined on a sphere, and show that the trajectory of any initial point that is not a fixed point isn't convergent. We study ergodicity of this $p$-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For $p\geq 3$ we prove that the dynamical systems are not ergodic. But for $p=2$ under some conditions the dynamical system may be ergodic.

math.DS↗

arXiv2022

Periodic points of a $p$-adic operator and their $p$-adic Gibbs measures

In this paper we investigate generalized Gibbs measure (GGM) for $p$-adic Hard-Core(HC) model with a countable set of spin values on a Cayley tree of order $k\geq 2$. This model is defined by $p$-adic parameters $λ_i$, $i\in \mathbb N$. We analyze $p$-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by $p$-adic version of Kolmogorov's theorem. We define $p$-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our $p$-adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters $p$, $k$ and $λ_i$ we find the number of translation-invariant and two-periodic GGMs for the $p$-adic HC model on the Cayley tree of order two.

math.FA↗

arXiv2019

$p$-Adic $(3,2)$-rational dynamical systems with three fixed points

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.

math.DS↗

arXiv2019

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 $x_2$, when $x_2$ is indifferent or attractor, respectively. When $x_2$ is repeller we find an open ball any point of which repelled from $x_2$. Moreover, we study relations between the sets $SI(x_1)$ and $SI(x_2)$ when $x_2$ is indifferent. For each $(2,2)$-rational function on $\mathcal{C}_p$ there are two points $\hat x_1=\hat x_1(f)$, $\hat x_2=\hat x_2(f)\in \mathcal{C}_p$ which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point $x_1$) containing some points such that the trajectories (under actions of $f$) of the points after a finite step come to $\hat x_1$ or $\hat x_2$. We study periodic orbits of the dynamical system and find an invariant set, which contains all periodic orbits. Moreover, we study ergodicity properties of the dynamical system on each invariant sphere. Under some conditions we show that the system is ergodic iff $p=2$.

math.DS↗

arXiv2018

$p$-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 containing the set of limit points. It is given all possible invariant spheres. We show that the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of $p$-adic numbers $Q_p$. Moreover some periodic orbits of the system are investigated.

math.DS↗

arXiv2018

$p$-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 $\hat x=\hat x(f)\in \mathbb C_p$ which is zero in its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point) containing some points that the trajectories (under actions of $f$) of the points after a finite step come to $\hat x$. For a class of $(3,1)$-rational functions defined on the set of $p$-adic numbers $\mathbb Q_p$ we study ergodicity properties of the corresponding dynamical systems. We show that if $p\geq 3$ then the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. For $p=2$, under some conditions we prove non ergodicity and show that there exists a sphere on which the dynamical system is ergodic. Finally, we give a characterization of periodic orbits and some uniformly local properties of the $(3.1)-$rational functions.

math.DS↗

arXiv2017

$p$-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 $(2,2)$-rational function on $\mathbb{C}_p$ there are two points $\hat x_1=\hat x_1(f)$, $\hat x_2=\hat x_2(f)\in \mathbb{C}_p$ which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point) containing some points such that the trajectories (under actions of $f$) of the points after a finite step come to $\hat x_1$ or $\hat x_2$. Moreover for a class of $(2,2)$-rational functions we study ergodicity properties of the dynamical systems on the set of $p$-adic numbers $Q_p$. For each such function we describe all possible invariant spheres. We show that the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure.

math.DS↗

arXiv2015

The dynamical system generated by the floor function $\lfloorλx\rfloor$

We investigate the dynamical system generated by the function $\lfloorλx\rfloor$ defined on $\mathbb R$ and with a parameter $λ\in \mathbb R$. For each given $m\in \mathbb N$ we show that there exists a region of values of $λ$, where the function has exactly $m$ fixed points (which are non-negative integers), also there is another region for $λ$, where there are exactly $m+1$ fixed points (which are non-positive integers). Moreover the full set $\mathbb Z$ of integer numbers is the set of fixed points iff $λ=1$. We show that depending on $λ$ and on the initial point $x$ the limit of the forward orbit of the dynamical system may be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbit or (iii) $\pm\infty$.

math.DS↗

arXiv2013

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 $x_0$ is a repeller point then the trajectory of an inner point from a ball $U_r(x_0)$ goes forward to the sphere $S_r(x_0)$. Once the trajectory reaches the sphere, in the next step it either goes back to the interior of $U_r(x_0)$ or stays in $S_r(x_0)$ for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of $U_r(x_0)$ it will stay (for all the rest of time) in the sphere (outside of $U_r(x_0)$) that it reached first.

math.DS↗

arXiv2011

$p$-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 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.

math.DS↗