Theoretical and Mathematical Physics, 170(3): 376–383 (2012)
A POLYNOMIAL p-ADIC DYNAMICAL SYSTEM F. M. Mukhamedov∗ and U. A. Rozikov† We completely describe the Siegel discs and attractors for the p-adic dynamical system f (x) = x2n+1 + axn+1 on the space of complex p-adic numbers.
Keywords: polynomial dynamical system, attractor, Siegel disc, p-adic number
1. Introduction The p-adic numbers were first introduced by the German mathematician Hensel; after their discovery, they were primarily considered a purely mathematical object of study. Beginning in the 1980s, various models described in the language of p-adic analysis have been actively studied. More precisely, models over the p-adic number field were considered because these numbers supposedly provide a more accurate and adequate description of the microworld. Various applications of these numbers to problems of theoretical physics [1]–[5], quantum mechanics [6], and many other branches of physics [7], [8] have been proposed. Studies in p-adic quantum mechanics stimulated research into p-adic dynamical systems (see, e.g., [9]– [12]). Some steps in this direction [9] indicate that even simple (monomial) discrete dynamical systems over the p-adic number field Qp exhibit a rather complex behavior depending essentially on the value of the prime p. Changing p, we transform attractors into Siegel discs and back. The number of cycles and the cycle lengths also depend on p [13]. Ergodic properties of such dynamical systems were considered in [14]. Applications of p-adic dynamical systems to some biological and physical systems were considered in [10], [11], [15], [16]. The basics of p-adic analysis and p-adic mathematical physics are given in [7], [17]. Analytic functions are known to play a fundamental role in complex analysis; rational functions play an analogous role in p-adic analysis [18], [19]. It is therefore natural to study rational function dynamics in p-adic analysis. On the other hand, p-adic dynamical systems are used when studying p-adic Gibbs measures [20]–[22]. Julia and Fatou sets for rational p-adic systems were studied in [23]–[25], where the analogue of the Sullivan theorem [26] for such dynamical systems was also proved. These advances have stimulated studies of components of Fatou sets for polynomial dynamical systems including attractors and Siegel discs [24], [25], [27]. Here, we study Siegel discs and attractors for a p-adic dynamical system of the form f (x) = x2n+1 + n+1 ax in Cp . We note that our investigation is substantially based on the p-adic calculus.
2. Preliminary information 2.1. p-Adic numbers. Here and hereafter, p is a fixed prime number. Let Qp be the p-adic number field, which is the completion of the rational number field Q with respect to the p-adic norm defined on ∗
International Islamic University Malaysia, Kuatan, Malaysia, e-mail:
[email protected].
†
Institute for Mathematics and Information Technologies, National Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 170, No. 3, pp. 448–456, March, 2012. Original article submitted May 1, 2011. 376
c 2012 Springer Science+Business Media, Inc. 0040-5779/12/1703-0376
Q as follows. Every rational number x = 0 can be written in the form x = pr (n/m), where the numbers n and m are not divisible by p. The p-adic norm of x is then |x|p = p−r . This norm satisfies the strong triangle inequality |x + y|p ≤ max{|x|p , |y|p }, which indicates the non-Archimedean property of the norm. This inequality implies that if |x|p = |y|,
then |x − y|p = max{|x|p , |y|p },
if |x|p = |y|p ,
then |x − y|p ≤ |x|p .
(1)
The field Qp is not algebraically complete; we let Qap denote its algebraic completion. By virtue of the Krull theorem [28], the norm in Qp admits a unique extension to Qap , which is itself non-Archimedean. We briefly recall the procedure for extending the norm. It suffices to extend the norm to a one-point algebraic extension. Let z be an algebraic number over the field Qp . We consider Qp (z), the minimal field containing both z and Qp . We can then regard Qp (z) as a vector space over Qp . Because z is algebraic, the space Qp (z) is finite-dimensional. We let {e1 , . . . , en } denote a basis in Qp (z). Every element x ∈ Qp (z) can be n represented in the form x = k=1 ξk ek . We define the norm of x as x1 = max |ξk |p . 1≤k≤n
We set M = maxi,j ei ej 1 and define the new norm x = M x1 . We define the extension of this norm on Qap as |x| = lim n xn , n→∞
and we conveniently let the same symbol | · |p denote the extended norm. We note that the field Qap is not complete with respect to this norm but the completion of the field Qap is algebraically complete [28]. We let Cp denote this completion and call it the field of complex p-adic numbers. For any a ∈ Cp and r > 0, we set Ur (a) = {x ∈ Cp : |x − a|p ≤ r}, Vr (a) = {x ∈ Cp : |x − a|p < r}, Sr (a) = {x ∈ Cp : |x − a|p = r}. Proofs of the following lemmas can be found in [9], [28]. Lemma 1. If a ∈ S1 (0), then S1 (0) \ V1 (a) ⊂ S1 (a). Lemma 2. Let Cnk = n! /k! (n − k)!, k ≤ n. Then |Cnk |p ≤ 1. For m ∈ N, we introduce the notation Γ(m) = {x ∈ Cp : xm = 1},
Γ=
∞
Γ(m) ,
m=1
Γm =
∞ j=1
j
Γ(m ) ,
Γu =
∞
Γm ,
m : (m,p)=1
and we let θj,k , j = 1, k, denote the kth root of unity, while θ1,k = 1. Here and hereafter, we let ( · , · ) denote the greatest common divisor, and the equality (m, p) = 1 therefore means that the numbers m and p are coprime. Lemma 3. We have the following statements: 377
1. Let y n = a, where a = θj,n−1 for some j = 1, n − 1 and y = a. If (n, p) = 1, then y ∈ S1 (a). 2. We have the inclusion Γu ⊂ S1 (1). 3. For any j = 1, pk − 1, we have the inequality |Cpjk |p ≤ 1/p. 4. We have the inclusion Γp ⊂ V1 (1). We say that a function f : Vr (a) → Cp is analytic if it can be represented in the form f (x) =
∞
fn (x − a)n ,
f n ∈ Cp ,
n=0
where we understand the convergence in the sense of the norm in the ball Vr (a). A more detailed exposition of analytic function theory can be found in [29]. 2.2. Dynamical systems in Cp . We briefly recall [27], [29] some known facts about a dynamical system (f, U ) on Cp , where f : x ∈ U → f (x) ∈ U is an analytic function and U = Vr (a) or U = Cp . Let f : U → U be an analytic function. We set f n (x) = f ◦ · · · ◦ f (x),
x ∈ U.
n
If f (x0 ) = x0 , then we call x0 a stable point of the function f . We say that a stable point x0 is attractive if there is a neighborhood V (x0 ) of x0 such that limn→∞ f n (y) = x0 for all y ∈ V (x0 ). If x0 is an attractive point, then we call the set
A(x0 ) = x ∈ Cp : f n (x) −→ x0 , n→∞
its basin of attraction. A stable point x0 is said to be repulsive if we have a neighborhood V (x0 ) of this point such that |f (x) − x0 |p > |x − x0 |p for x ∈ V (x0 ), x = x0 . Let x0 be a stable point of a function f (x). We call a ball Vr (x0 ) (in U ) a Siegel disc if every sphere Sρ (x0 ), ρ < r, is an invariant sphere with respect to f (x), i.e., if x ∈ Sρ (x0 ), then all iterations of this point lie inside the same sphere, f n (x) ∈ Sρ (x0 ) for all n = 1, 2 . . . . The union of all Siegel discs centered at x0 is called the maximum Siegel disc, denoted by SI(x0 ). Remark 1 [9]. In complex geometry, the disc center is uniquely determined by the disc, and different stable points cannot share the same Siegel disc. In the non-Archimedean case, a disc center is any point belonging to this disc, and the same Siegel disc can in principle correspond to different stable points, which distinguishes the non-Archimedean case from the standard case. Let x0 be a stable point of an analytic function f (x), and let λ = df (x0 )/dx. The point x0 is attractive if 0 ≤ |λ|p < 1, a saddle point if |λ|p = 1, and repulsive if |λ|p > 1. Theorem 1 [9]. Let x0 be a stable point of an analytic function f : U → U . Then we have the following statements: 1. If x0 is an attractive stable point of the function f , then it is an attractive point of the dynamical system (f, U ). If a number r > 0 satisfies the inequality 1 dn f n−1 (x )
