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0THE DYNAMICAL SYSTEM OF (2,1) -RATIONAL FUNCTION WITH UNIQUE MULTIPLE FIXED POINT
1Jurayev Shuxrat Maxamadaliyevich, 2Ahmadaliyev Shohruh Bahromjon O'g'li
1Andijan State University, 2Namangan State University https://doi.org/10.5281/zenodo.13894665
x2 ax
Annotatsiya. Ushbu maqolada biz f (x) =- ko'rinishdagi ratsional funksiyaning
bx + a
diskret vaqtli p -adik dinamik sistemasini o'rgandik. O'rganilgan ratsional funksiya yagona x = 0 qo'zg'almas nuqtaga ega hamda uning betaraf qo'zg'almas nuqta ekanligini ko'rsatdik. Bundan tashqari, b parametrning turli qiymatlarida maksimal Zigel disklari topdik.
Kalit so'zlar: p -adik son, qo'zg'almas nuqta, itaruvchi, tortuvchi, betaraf, Zigel diski, maksimal Zigel diski.
Aннотация. В данной статье мы изучили дискретную систему динамики на p -
2
адических числах для рациональной функции вида f (x) =-. Мы показали, что
bx + a
рациональная функция имеет единственную неподвижную точку при x = 0 и доказали, что эта неподвижная точка является нейтральной. Кроме того, мы нашли максимальные диски Зигеля для различных значений параметра b .
Annotation. In this article, we studied the discrete-time p -adic dynamic system of the
x 2 + ax
rational function of the form f (x) =-. We demonstrated that the rational function has a
bx + a
unique fixed point at x = 0 and showed that this fixed point is neutral. Additionally, we found the maximal Siegel disks for different values of the parameter b .
Key words: p -adic number, fixed point, attractor, repeller , indifferent, Siegel disk, maximum Siegel disk.
1. p -adic numbers. Let □ be the field of rational numbers. The greatest common divisor of the positive integers n and m is denoted by (n, m). Every rational number x И 0 can be
n
represented in the form x = pr —, where r, n e □ , m is a positive integer, (p, n) = 1, (p, m) = 1
m
and pis a fixed prime number.
The p -adic norm of x is given by
_j p-, for x И 0, Xp =[0, for x = 0.
It has the following properties:
1. | x \p > 0 and | x \p = 0 if and only if x = 0,
2. \ xJ \p =\x \p \ y \p,
3. The strong triangle inequality: \ x + y \ < max{\x y \ }, ЗЛ) if \ x \pИу \p, then \ x + y \p = max{\x \p,\y \p},
3.2) if \ x \p =\y \p, then \ x + y \p<\ x \p ,
This is a non-Archimedean one. The completion of □ with respect to p -adic norm defines the p -adic field which is denoted by □ (see [1]). The algebraic completion of □ is denoted by □ , and it is called complex p -adic numbers.
For any a eD and r > 0 denote:
£/.(a) = {xeD p :\x-a\p<r}, Vr(a) = {xe□ p :\x-a\p<r}, Sr(a) = {x<=U p:\x-a\p = r}
A function /: Ur (a) —> □ is said to be analytic if it can be represented by:
/(*) = £/„(*-")", /„e Up,
n=0
which converges uniformly on the ball Ur (a)
2. Dynamical systems in □ p. Recall some known facts concerning dynamical systems (f,U) in □ where / :x <eU —> f(x)eU is an analytic function and U = Ur(a) or □ (see, for example, [2,3]).
Now let /: U —> U be an analytic function. Denote /"(x) = f°- ■ ■ °f {x) (n times).
• If f (x0) = x0 then x0 is called a fixed point. The set of all fixed points of f is denoted by Fix( f). A fixed point x0 is called an attractor if there exists a neighborhood U(x0) of x0 such that for all points x e U(x0), it holds lim fn (x) = x0.
• A fixed point x0 is called a repeller if there exists a neighborhood U(x0) such that | f (x) - x0 | >| x - x0 \ for x e U(x0), x ^ x0 .
Let x0 be a fixed point of a function f (x) ). Put X = f '(x0) . The point x0 is attractive if 0 <| X | < 1, indifferent if | X\ = 1, and repelling if | X\ > 1.
The ball Ur (x0) (contained in V) is said to be a Siegel disk if each sphere £ (x0), p < r is an invariant sphere of f (x), i.e., if x e £p(x0), then all iterated points fn(x) e £p(x0) for all n = 1,2,3,.... The union of all Siegel disks with the center at x0 is said to be a maximum Siegel disk and is denoted by SI (x0).
3. (2,1)-Rational p -adic dynamical systems. In this paper we consider the dynamical system associated with the (2,1) -rational function /:□—>•□ defined by
2
f (x) = x-+ax, (1)
bx + a
where x ^ —b.
a
The function (1) has unique fixed point x = 0 . Thus we study dynamical system (/, Cp) with f given by (1). For (1) we have
f'(xo) = f '(0) = 1
i.e., the point x0 is an indifferent point for (1).
We get the following main result:
Theorem. The following statement holds:
SI(0)=
U (0) if | b | < 1,
\a\p V J ' 'p
U.a. (0) if | b | > 1.
REFERENSCES:
1. F.Q. Gouvea, p — Adic Numbers. An Introduction, Springer-Verlag, Berlin Heidelberg, New York, second, 1997.
2. U.A. Rozikov, I.A. Sattarov. On a non-linear p-adic dynamical system. p —Adic Numbers, Ultrametric analysis and Applications. 2014(6), 53-64.
I.A. Sattarov. p — adic (3,2)-rational dynamical systems. p —Adic Numbers, Ultrametric analysis and Applications. 2015(7), 39-55.
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