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0INTERNATIONAL SCIENTIFIC-PRACTICAL CONFERENCE INNOVATIVE DEVELOPMENT OF SCIENCE AND EDUCATION: NEW APPROACH AND RESEARCH
OCTOBER 5, 2024
P-ADIC DYNAMICAL SYSTEMS OF THE FUNCTION AX-K
Abdurahmonova M.SH
Namangan State University https://doi.org/10.5281/zenodo.13894966
Abstract. In this talk we consider the dynamical systems generated by the function /(x) = in the field of p-adic numbers Q.
To define a dynamical system we consider a function / : x E U ^ /(x) E t/. For x E U denote by /n(x) the «-fold composition of f with itself (i.e. n time iteration off to x):
/»(*) = /(/(/( .. .(/(X)) ) . .. ).
For arbitrary given x0 E U and / : U ^ U the discrete-time dynamical system (also called the trajectory) of x0 is the sequence of points:
Xo, Xi = /(Xo), X2 = /(Xi) = /2(Xo), X3 = /(X2) = /3(xo),... xn = /n(xo),...
This expression is called the orbit of the point xo under the influence of the function f(x). Where point xo is called the origin of the orbit.
A point x E U is called a fixed point for f if /(x) = x. The point x is a periodic point of period m if /m(x) = x .The least positive m for which /m(x) = x is called the prime period of x . A fixed point xois called an attractor if there exist a neighborhood U(xo) of xo such that for all points x E U(xo) it holds
lim /n(x) = xo .If xo is an attractor then its basin of attraction is
n^œ
^(xo)=(xE R : /n(x) ^ xo , n ^ œ).
A fixed point xo is called repeller if there exists a neighborhood t/(xo) of xo such that |/(x) — xo|p > |x — xo|p for xE ^(xo), x * xo. Let xo be a fixed point of a function /(x). Put A = /'(xo).The point xois attractive if 0 < |A|p <1,indifferent if |A|p =1, and repellingif |A|p >1.
The ball t/r(xo) (contained in V) is said to be a Siegel disk if each sphere Sp(xo), p<r is an invariant sphere of /(x), i.e. if x E Sp(xo), then all iterated points /n(x) E 5p(xo)
for all n=1,2,... . The union of all Siegel disks with the center at xo is called a maximum Siegel disk and is denoted by SI(xo).
In this work we consider the discrete dynamical system associated with the function / : Qp ^ Qp defined by
The fixed points of this function are solutions of the equation xK+1 = a in Qp. The work
/(x) = a * 0, flEQp, fcEi, where x * x = 0. (1)
ck+i ,fc _
provides detailed information about the solutions of the equation xK = a in the field Qp. Using these data, we obtain the following results. Define the set
5o/p(xfc+1 — fl) = {(EFp : = a(modp)},a E Zp, where Fp := {0,1,2, .„,p — 1) and Zp := (x E Q : |x|p = 1). We also denote the number of elements of the set 5o/p(xfc+1 — a) by fcp. Note that for any number fcEi, there are numbers m E N and s E Z+ such that k = mps, (m, p) = 1 is valid. Let
a = ao + a1 • p + a2 • p2 + a3 • p3 + _ (2).
Theorem 1. Let p > 3 and k + 1 = mps .If the canonical expansion of the number a E Zp is (2), then the following assertions are valid:
INTERNATIONAL SCIENTIFIC-PRACTICAL CONFERENCE INNOVATIVE DEVELOPMENT OF SCIENCE AND EDUCATION: NEW APPROACH AND RESEARCH
OCTOBER 5, 2024
(i) If Solp(xk+1 — a) = 0 then the equation xk+1 = a hasn't solution;
(ii) Let SoIp(xk+1 — a) & 0. Then
(ii)1) if la — aP lp > p-s, then the equation xk+1 = a hasn't solution;
(ii)2) if la — ap lp < p-s, then the number of solutions of the equation xk+1 = a
is equal to Kp.
Theorem 2. Let p > 3 and let a E Then the following assertions hold:
(i) if \klp<|a — 1lp, then (1) has no solution;
(ii) if \k|p>|a — 1|p, then (1) has exactly Kp solutions x^. E B^i)^ E {1,2, ...,Kp}, where ^ E Solp(xk+1 — a).
It can be seen from this theorem that the number of fixed points of the function (1) may be equal to kp.
Theorem 3. Let p > 3,k + 1 = mps,a E lp and \ a — ap lp < p-s. Then the fixed points of the function (1) is an attractive for plk and an indifferent for p \ k. References
1. Mukhamedov F., Khakimov O. On equation xk = a over Qp and its applications. IzvestiyaMath. RAN. 2020. 84 pp. 348-360.
2. Rozikov U.A. p-Adic dynamical system of the function ax~2. doi.org/10.48550/arXiv.2101.05750
