p -adic (2,1) -rational dynamical systems
aa r X i v : . [ m a t h . D S ] N ov p -ADIC (2 , -RATIONAL DYNAMICAL SYSTEMS ALBEVERIO S, ROZIKOV U.A., SATTAROV I.A.
Abstract.
We investigate the behavior of the trajectory of an arbitrary (2 , p -adic dynamical system in a complex p -adic field C p . We study, in particular, Siegeldisks and attractors for such dynamical systems. The set of fixed points of the (2 , p -adic dynamical systemhas a 2-periodic cycle x , x which only can be either an attracting or an indifferent one.If it is attracting then it attracts each trajectory which starts from an element of a ballof radius r = | x − x | p with the center at x or at x . If the 2-periodic cycle is anindifferent one, then in each step the above mentioned balls transfer to each other. Allthe other spheres with radius > r and the center at x and x are invariant independentlyof the attractiveness of the cycle.(ii) In the case where the fixed point x is unique we prove that if the point is attractingthen there exists δ >
0, such that the basin of attraction for x is the ball of radius δ andthe center at x and any sphere with radius ≥ δ is invariant. If x is an indifferent pointthen all spheres with the center at x are invariant. If x is a repelling point then thereexits δ >
0, such that the trajectory which starts at an element of the ball of radius δ with the center in x leaves this ball, whereas any sphere with radius ≥ δ is invariant.(iii) In case of existence of two fixed points, the p -adic dynamical system has a veryrich behavior: we show that Siegel disks may either coincide or be disjoint for differentfixed points of the dynamical system. Besides, we find the basin of the attractor ofthe system. Varying the parameters it is proven that there exists an integer k ≥ S r ( x i ) , . . . , S r k ( x i ) such that the limiting trajectory will be periodicallytraveling the spheres S r j . For some values of the parameters there are trajectories whichgo arbitrary far from the fixed points. Introduction
In this paper we will state some results concerning discrete dynamical systems definedover the complex p -adic field C p . The interest in such systems and in the ways they canbe applied has been rapidly increasing during the last couple of decades (see, e.g., [31]and the references therein). The p -adic numbers were first introduced by the Germanmathematician K.Hensel. For about a century after the discovery of p -adic numbers,they were mainly considered objects of pure mathematics. Beginning with 1980’s variousmodels described in the language of p -adic analysis have been actively studied. Moreprecisely, models over the field of p -adic numbers have been considered which, based on theconception that p -adic numbers might provide a more exact and more adequate descriptionof micro-world phenomena. Numerous applications of these numbers to theoretical physics Mathematics Subject Classification.
Key words and phrases.
Rational dynamical sustems; attractors; Siegel disk; complex p -adic field. have been proposed in papers [4], [13], [23], [34], [35] to quantum mechanics [18], to p -adic- valued physical observable [18] and many others [19], [33].The study of p -adic dynamical systems arises in Diophantine geometry in the construc-tions of canonical heights, used for counting rational points on algebraic vertices overa number field, as in [7]. In [20], [32] p -adic field have arisen in physics in the theoryof superstrings, promoting questions about their dynamics. Also some applications of p -adic dynamical systems to some biological, physical systems has been proposed in [1]-[3], [8], [20], [21]. Other studies of non-Archimedean dynamics in the neighborhood of aperiodic and of the counting of periodic points over global fields using local fields appearin [14], [22], [28]. It is known that the analytic functions play important role in complexanalysis. In the p -adic analysis the rational functions play a similar role to the analyticfunctions in complex analysis [30]. Therefore, naturally one arises a question to study thedynamics of these functions in the p -adic analysis. On the hand, such p -adic dynamicalsystems appear while studying p -adic Gibbs measures [9], [10], [15], [25]- [27]. In [5], [6]dynamics on the Fatou set of a rational function defined over some finite extension of Q p have been studied, besides, an analogue of Sullivan’s no wandering domains theorem for p -adic rational functions which have no wild recurrent Julia critical points were proved.In [3] the behavior of a p -adic dynamical system f ( x ) = x n in the fields of p -adic numbers Q p and complex p -adic numbers C p was investigated. Some ergodic properties of thatdynamical system have been considered in [12].In [24] the behavior of the trajectory of a rational p -adic dynamical system in complex p -adic filed C p is studied. It is studied Siegel disks and attractors of such dynamicalsystems. It is shown that Siegel disks may either coincide or be disjoint for different fixedpoints. Besides, the basin of the attractor of the rational dynamical system is found. Itis proved that such kind of dynamical system is not ergodic on a unit sphere with respectto the Haar measure.The base of p -adic analysis, p -adic mathematical physics are explained in [11], [17], [33].In this paper we investigate the behavior of trajectory of an arbitrary (2 , p -adic dynamical system in complex p -adic field C p . The paper is organized as follows:in Section 2 we give some preliminaries. Section 3 contains the definition of the conceptof a (2 , p -adic dynamical system whichhas a unique fixed point x . We prove that if the point is attracting then there exits δ >
0, such that the basin of attraction for x is the ball of radius δ and the center at x and any sphere with radius ≥ δ is invariant. If x is an indifferent point then all sphereswith the center at x are invariant. If x is a repelling point then there exits δ >
0, suchthat the trajectory which starts at an element of the ball of radius δ with center at x leaves this ball, whereas any sphere with radius ≥ δ is invariant. Section 5 contains resultsconcerning the p -adic dynamical systems which have no fixed point. We show that such p -adic dynamical systems have a 2-periodic cycle x , x which only can be an attractingor an indifferent one. If it is attracting then it attracts each trajectory which starts froman element of a ball of radius r = | x − x | p with the center at x or at x . If the 2-periodic cycle is an indifferent one then in each step the above mentioned balls transferto each other. All the other spheres with radius > r and the center at x and x areinvariant independently on the attractiveness of the cycle. The last section is devoted tothe case of existence of two fixed points. We show, in particular, that Siegel disks mayeither coincide or be disjoint for different fixed points of the dynamical system. Besides, -ADIC (2 , we find the basin of the attractor of the system. Varying the parameters we prove thatthere exists k ≥
2, and spheres S r ( x i ) , . . . , S r k ( x i ) such that the limiting trajectory willbe periodically traveling the spheres S r j ( x i ). For some values of the parameters there aretrajectories which go arbitrary far from the fixed points.2. Preliminaries p -adic numbers. Let Q be the field of rational numbers. The greatest commondivisor of the positive integers n and m is denotes by ( n, m ). Every rational number x = 0can be represented in the form x = p r nm , where r, n ∈ Z , m is a positive integer, ( p, n ) = 1,( p, m ) = 1 and p is a fixed prime number.The p -adic norm of x is given by | x | p = ( p − r , for x = 0 , , for x = 0 . It has the following properties:1) | x | p ≥ | x | p = 0 if and only if x = 0,2) | xy | p = | x | p | y | p ,3) the strong triangle inequality holds | x + y | p ≤ max {| x | p , | y | p } , | x | p = | y | p then | x + y | p = max {| x | p , | y | p } ,3.2) if | x | p = | y | p then | x + y | p ≤ | x | p .Thus | x | p is a non-Archimedean norm.The completion of Q with respect to the p -adic norm defines the p -adic field which isdenoted by Q p .The well-known Ostrovsky’s theorem asserts that the norms | x | = | x | ∞ and | x | p , p ∈ P,(where P = { , , ... } denotes the set of prime numbers) exhaust all nonequivalentnorms on Q (see [17]). Any p -adic number x = 0 can be uniquely represented by thecanonical series, convergence in the | x | p -norm: x = p γ ( x ) ( x + x p + x p + ... ) , (2 . γ = γ ( x ) ∈ Z and x j are integers, 0 ≤ x j ≤ p − x > j ∈ N = { , , , ... } (seefor more details [11], [17]). Observe that in this case | x | p = p − γ ( x ) .The algebraic completion of Q p which is complete with respect to | · | p is denoted by C p and it is called the field of complex p -adic numbers . For any a ∈ C p and r > U r ( a ) = { x ∈ C p : | x − a | p ≤ r } , V r ( a ) = { x ∈ C p : | x − a | p < r } ,S r ( a ) = { x ∈ C p : | x − a | p = r } . A function f : U r ( a ) → C p is said to be analytic if it can be represented by f ( x ) = ∞ X n =0 f n ( x − a ) n , f n ∈ C p , which converges uniformly on the ball U r ( a ). ALBEVERIO S, ROZIKOV U.A., SATTAROV I.A.
Theorem 2.2. [33] Let f ( x ) be an analytic function on a ball U r ( a ) and f ′ ( a ) = 0 , | f ′ ( a ) | p = p n . Then there exists a ball U ρ ( a ) , with ρ < r , such that f is one-to-onemap from U ρ ( a ) into U ρ + n ( b ) with b = f ( a ) , and the inverse function g ( y ) is an analyticfunction on U ρ + n ( b ) and the following equality holds g ′ ( b ) = 1 f ′ ( a ) . Dynamical systems in C p . In this section we recall some known facts concerningdynamical systems ( f, U ) in C p , where f : x ∈ U → f ( x ) ∈ U is an analytic function and U = U r ( a ) or C p .Now let f : U → U be an analytic function. Denote x n = f n ( x ), where x ∈ U and f n ( x ) = f ◦ · · · ◦ f | {z } n ( x ).Let us first recall some the standard terminology of the theory of dynamical systems(see for example [29]). If f ( x ) = x then x is called a fixed point . The set of all fixedpoints of f is denoted by Fix( f ). A fixed point x is called an attractor if there exists aneighborhood V ( x ) of x such that for all points y ∈ V ( x ) it holds that lim n →∞ y n = x .If x is an attractor then its basin of attraction is A ( x ) = { y ∈ C p : y n → x , n → ∞} . A fixed point x is called repeller if there exists a neighborhood V ( x ) of x such that | f ( x ) − x | p > | x − x | p for x ∈ V ( x ), x = x . Let x be a fixed point of a function f ( x ). The ball V r ( x ) (contained in U ) is said to be a Siegel disk if each sphere S ρ ( x ), ρ < r is an invariant sphere for f ( x ), i.e. if x ∈ S ρ ( x ) then for all iterated points we have x n ∈ S ρ ( x ), for all n ∈ N . The union of all Siegel disks with the center at x is said to amaximum Siegel disk and is denoted by SI ( x ).In complex geometry, the center of a disk is uniquely determined by the disk, anddifferent fixed points cannot have the same Siegel disks. In non-Archimedean geometry, acenter of a disk is a point which belongs to the disk. Therefore, different fixed points mayhave the same Siegel disk [3].Let x be a fixed point of an analytic function f ( x ). Put λ = ddx f ( x ) . The point x is attractive if 0 ≤ | λ | p < indifferent if | λ | p = 1, and repelling if | λ | p > Theorem 2.4. [3] Let x be a fixed point of an analytic function f : U → U . Thefollowing assertions hold if x is an attractive point of f , then it is an attractor of the dynamical system ( f, U ) . If r > satisfies the inequality q = max ≤ n< ∞ (cid:12)(cid:12)(cid:12)(cid:12) n ! d n fdx n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p r n − < . and U r ( x ) ⊂ U then U r ( x ) ⊂ A ( x ) ; -ADIC (2 , if x is an indifferent point of f then it is the center of a Siegel disk. If r satisfiesthe inequality s = max ≤ n< ∞ (cid:12)(cid:12)(cid:12)(cid:12) n ! d n fdx n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p r n − < | f ′ ( x ) | p (2 . and U r ( x ) ⊂ U then U r ( x ) ⊂ SI ( x ) ; if x is a repelling point of f then x is a repeller of the dynamical system ( f, U ) .
3. (2 , -Rational p -adic dynamical systems A function is called an ( n, m )-rational function if and only if it can be written in theform f ( x ) = P n ( x ) Q m ( x ) , where P n ( x ) and Q m ( x ) are polynomial functions with degree n and m respectively ( Q m ( x ) is not the zero polynomial).In this paper we consider the dynamical system associated with the (2 , f : C p → C p defined by f ( x ) = x + ax + bcx + d , a, b, c, d ∈ C p , c = 0 , d − acd + bc = 0 , (3.1)where x = ˆ x = − dc .Note that any (2 , a = b = 0 then the function coincides with the function considered in [16]. But theauthors of [16] did not consider the case c = 1 , d = 0 in which case the function has nofixed point. In this paper we consider the following cases:1) c = 1, a = d . In this case the function f has a unique fixed point x = bd − a .2) c = 1, a = d , b = 0. In this case there is no fixed point.3) c = 1, a = d , b = 0. In this case f ( x ) = x , i.e. the function f is the id-function.4) c = 1. There are two fixed points: x , = a − d ± p ( a − d ) + 4( c − b c − . (3.2)The function f can be written in the following form f ( x ) = 1 c c x + ac − dc + d − acd + bc x + dc ! . Using this representation we get the following formulas: f ′ ( x ) = 1 c c − d − acd + bc (cid:0) x + dc (cid:1) ! , (3.3) d n d n x f ( x ) = ( − n n ! d − acd + bc c (cid:0) x + dc (cid:1) n +1 , n ≥ . (3.4) ALBEVERIO S, ROZIKOV U.A., SATTAROV I.A. The case of a unique fixed point
In this section we consider the case c = 1 and a = d . As mentioned above in this casethere is a unique fixed point: x = bd − a .Denote P = { x ∈ C p : ∃ n ∈ N ∪ { } , f n ( x ) = ˆ x } , for c = 1 . For c = 1 we have f ′ ( x ) = ad + b − a d − ad + b . (4.1)4.1. The case | f ′ ( x ) | p = 1 . If | f ′ ( x ) | p = 1 then by Theorem 2.4 the point x is anindifferent point and it is the center of a Siegel disk. Theorem 4.2. If c = 1 , | f ′ ( x ) | p = 1 then SI ( x ) = C p \ P .Proof. Denote δ = | x + d | p . We shall prove that f ( S r ( x ) \ P ) ⊂ S r ( x ) for any r > y ∈ S r ( x ) \ P , i.e., y = x + γ , | γ | p = r . We have | f ( y ) − x | p = r · (cid:12)(cid:12)(cid:12)(cid:12) x + 2 dx + ad − b + γ ( x + d )( x + d ) + γ ( x + d ) (cid:12)(cid:12)(cid:12)(cid:12) p = r · (cid:12)(cid:12)(cid:12) f ′ ( x ) + γx + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) γx + d (cid:12)(cid:12)(cid:12) p . (4.2)For r = δ using property 3.1) of the p -adic norm we get | f ( y ) − x | p = r , i.e. f ( y ) ∈ S r ( x ). In case r = δ one can use the p -adic version of the inverse function given byTheorem 2.2 and show that f ( S δ ( x ) \ P ) ⊂ S δ ( x ). Indeed, if we assume that thereexists x ∈ S δ ( x ) such that y = f ( x ) / ∈ S δ ( x ) then this by the inverse function theoremgives the following contradiction: δ = | x − x | p = | f − ( y ) − x | p = (cid:12)(cid:12)(cid:12)(cid:12) f ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p | y − x | p = | y − x | p = δ. (cid:3) The case | f ′ ( x ) | p < . For | f ′ ( x ) | p < x is an attractive point of f . Theorem 4.4. If c = 1 , | f ′ ( x ) | p < then (i) for any µ > δ the sphere S µ ( x ) is invariant with respect to f , i.e. f ( S µ ( x )) ⊂ S µ ( x ) . (ii) V δ ( x ) ⊂ A ( x ) .Proof. (i) Take an arbitrary y ∈ S µ ( x ), i.e., y = x + γ , | γ | p = µ . We have | f ( y ) − x | p = µ · (cid:12)(cid:12)(cid:12) f ′ ( x ) + γx + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) γx + d (cid:12)(cid:12)(cid:12) p . (4.3)Using µ > δ and property 3.1) of the p -adic norm we get | f ( y ) − x | p = µ , i.e. f ( y ) ∈ S µ ( x ). -ADIC (2 , (ii) We shall use Theorem 2.4. By formula (3.4), for c = 1, we have q = max ≤ n< ∞ (cid:12)(cid:12)(cid:12)(cid:12) n ! · d n fdx n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p r n − = max ≤ n< ∞ (cid:12)(cid:12)(cid:12)(cid:12) d − ad + b ( x + d ) n +1 (cid:12)(cid:12)(cid:12)(cid:12) p r n − =max ≤ n< ∞ (cid:12)(cid:12)(cid:12)(cid:12) d − ad + b ( x + d ) (cid:12)(cid:12)(cid:12)(cid:12) p (cid:18) r | x + d | p (cid:19) n − = max ≤ n< ∞ (cid:12)(cid:12) − f ′ ( x ) (cid:12)(cid:12) p (cid:18) r | x + d | p (cid:19) n − =max ≤ n< ∞ (cid:18) r | x + d | p (cid:19) n − < , if r < δ. Thus by Theorem 2.4 we get V δ ( x ) ⊂ A ( x ). (cid:3) Hence we proved that all elements of S µ ( x ) for µ < δ are points of the basin ofattraction A ( x ), and all spheres S µ ( x ) for µ > δ are invariant. Moreover, since on S µ ( x ) for µ > δ there is no a fixed point of f the trajectory x n = f n ( x ) does notconverge for any x ∈ S µ ( x ), for µ > δ . Now it remains to study the dynamical systemfor x ∈ S δ ( x ). Lemma 4.5. P ⊂ S δ ( x ) .Proof. First we note that ˆ x ∈ S δ ( x ). Indeed, for c = 1 we have | ˆ x − x | p = | − d − x | p = | x + d | p = δ. Now take an arbitrary x ∈ P , x = ˆ x then x is not in S µ ( x ), ∀ µ > δ since the spheres areinvariant with respect to f , so there is no n with f n ( x ) = ˆ x . Moreover, x / ∈ V δ ( x ), since V δ ( x ) is subset of the attractor. Thus the only possibility is that x ∈ S δ ( x ). (cid:3) Theorem 4.6. If c = 1 , | f ′ ( x ) | p < and x ∈ S δ ( x ) \ P then there exist the followingtwo possibilities: There exists k ∈ N and µ k > δ such that f m ( x ) ∈ S µ k ( x ) , for any m ≥ k . The trajectory { f k ( x ) , k ≥ } is a subset of S δ ( x ) .Proof. Take x ∈ S δ ( x ) \ P then x = x + γ with | γ | p = δ . We have | f ( y ) − x | p = δ · (cid:12)(cid:12)(cid:12) f ′ ( x ) + γx + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) γx + d (cid:12)(cid:12)(cid:12) p = δ (cid:12)(cid:12)(cid:12) γx + d (cid:12)(cid:12)(cid:12) p ≥ δ. (4.4)If | f ( x ) − x | p > δ then there is µ > δ such that f ( x ) ∈ S µ ( x ). So in this case k = 1.If | f ( x ) − x | p = δ then we consider the following equality | f ( x ) − x | p = | f ( x ) − x | p (cid:12)(cid:12)(cid:12) f ′ ( x ) + f ( x ) − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) f ( x ) − x x + d (cid:12)(cid:12)(cid:12) p = δ | f ( x ) + d | p . (4.5)Since f ( x ) ∈ S δ ( x ) we have f ( x ) = x + γ , with | γ | p = δ . From (4.5) we get | f ( x ) − x | p = δ | γ + ( x + d ) | p ≥ δ. ALBEVERIO S, ROZIKOV U.A., SATTAROV I.A.
Now, if | f ( x ) − x | p > δ then there is µ > δ such that f ( x ) ∈ S µ ( x ). So in this case k = 2. If | f ( x ) − x | p = δ then we can continue the argument and get the followingequality | f k ( x ) − x | p = δ | f k − ( x ) + d | p . Hence in each step we may have two possibilities: | f k ( x ) − x | p = δ or | f k ( x ) − x | p > δ .In case | f k ( x ) − x | p > δ there exists µ k such that f k ( x ) ∈ S µ k ( x ), and since S µ k ( x ) isan invariant with respect to f we get f m ( x ) ∈ S µ k ( x ) for any m ≥ k . If | f k ( x ) − x | p = δ for any k ∈ N then { f k ( x ) , k ≥ } ⊂ S δ ( x ). (cid:3) From Theorems 4.2-4.6 we get immediately the following
Corollary 4.7. A ( x ) = V δ ( x ) . The case | f ′ ( x ) | p > . If | f ′ ( x ) | p > x is a repeller of thedynamical system. Theorem 4.9. If c = 1 , | f ′ ( x ) | p = q > and x ∈ S µ ( x ) \P then the following propertieshold (a) If µ < δ then (a.1) If µq m = δ for all m ∈ N then there exists k ∈ N such that µq k > δ and f n ( x ) ∈ S µq n ( x ) , n = 1 , . . . , k ; f k +1 ( x ) ∈ S δq ( x ); f k +2 ( x ) ∈ S ν ( x ) , where ν ≤ δq, i.e. the trajectory after k steps leaves U δ ( x ) . (a.2) If µq m = δ for some m ∈ N then f n ( x ) ∈ S δqm − n ( x ) , n = 1 , . . . , m ; f m +1 ( x ) ∈ S ν ( x ) , where ν ≥ δq. (b) If µ > δ then (b.1) If µ > δq then f ( x ) ∈ S µ ( x ) ; (b.2) If µ < δq then f ( x ) ∈ S δq ( x ) ; (b.3) If µ = δq then f ( x ) ∈ S ν ( x ) where ν ≤ δq .Proof. (a.1) For x ∈ S µ ( x ) we have | f ( x ) − x | p = µ · (cid:12)(cid:12)(cid:12) f ′ ( x ) + x − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) x − x x + d (cid:12)(cid:12)(cid:12) p = µq. (4.6)If µq > δ then k = 1. If µq < δ then we take µ = µq and get | f ( x ) − x | p = µ · (cid:12)(cid:12)(cid:12) f ′ ( x ) + f ( x ) − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) f ( x ) − x x + d (cid:12)(cid:12)(cid:12) p = µ q = µq . If µq > δ then k = 2. Otherwise we take µ = µq and repeat the argument. Since q > k ∈ N such that µq k − < δ , µq k > δ and | f k ( x ) − x | p = µq k . -ADIC (2 , Since 1 < µq k δ < q , using property 3.1) we obtain | f k +1 ( x ) − x | p = µq k · (cid:12)(cid:12)(cid:12) f ′ ( x ) + f k ( x ) − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) f k ( x ) − x x + d (cid:12)(cid:12)(cid:12) p = µq k · q µq k δ = δq. For k + 2 we have | f k +2 ( x ) − x | p = δq · (cid:12)(cid:12)(cid:12) f ′ ( x ) + f k +1 ( x ) − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) f k +1 ( x ) − x x + d (cid:12)(cid:12)(cid:12) p ≤ δq · qq = δq. Thus there is ν ≤ δq such that f k +2 ( x ) ∈ S ν ( x ).(a.2) This proof up to the step m similar to the proof of (a.1). Since | f m ( x ) − x | p = δ ,for m + 1 we have | f m +1 ( x ) − x | p = δ · (cid:12)(cid:12)(cid:12) f ′ ( x ) + f m ( x ) − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) f m ( x ) − x x + d (cid:12)(cid:12)(cid:12) p ≥ δq. (b.1) For x ∈ S µ ( x ) using conditions of the assertion (b.1) we get | f ( x ) − x | p = µ · (cid:12)(cid:12)(cid:12) f ′ ( x ) + x − x x + d (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) x − x x + d (cid:12)(cid:12)(cid:12) p = µ · µ/qµ/q = µ. The proofs of (b.2) and (b.3) are similar to the above proofs. (cid:3)
Remark 4.10.
Theorem 4.9 gives the following character of the dynamical system when x is a repeller point: the trajectory of a point from the inner of U δq ( x ) goes forward tothe sphere S δq as soon as the trajectory reaches the sphere, in the next step, it may goback to the inner of U δq ( x ) or stay in S δq for some time and then go back to the inner of U δq ( x ) . As soon as the trajectory goes outside of U δq ( x ) it will stay (all the rest time)in the sphere (outside of U δq ( x ) ) where first it came. The case where there is no fixed point
In this section we consider the case c = 1, a = d , b = 0. In this case the function f ( x ) = x + ax + bx + a has no fixed point. In such a case it will be interesting to find periodicpoints of f . Let us consider 2-periodic points, i.e. consider the equation g ( x ) ≡ f ( f ( x )) = x + bx + a + b ( x + a )( x + a ) + b = x. (5.1)This equation is equivalent to ( x + a ) = − b , which has two solutions t , = − a ± p − b/ C p . It is a surprise that g ′ ( t ) = g ′ ( t ) = 9, i.e. the value does not depend on parameters a and b . Thus we have | g ′ ( t ) | p = | g ′ ( t ) | p = (cid:26) , if p = 3 , / , if p = 3 . (5.2)Note that the function g (see (5.1)) is defined in C p \ { ˆ x = − a, ˆˆ x ± } , where ˆˆ x ± = − a ± √− b are the solutions to the equation f ( x ) = ˆ x .Denote P = { x ∈ C p : ∃ n ∈ N , such that f n ( x ) ∈ { ˆ x, ˆˆ x − , ˆˆ x + }} ,h = | t + a | p = | t + a | p . The following equalities are obvious: | t − t | p = h, p = 2; | ˆ x − t | p = | ˆ x − t | p = h. Case p = 3 . In this case each fixed point t , t of g is an indifferent point and is thecenter of a Siegel disk. Theorem 5.2. If p = 3 then f ( S r ( t ) \ P ) ⊆ S r ( t ) , f ( S r ( t ) \ P ) ⊆ S r ( t ) , for any r > .Proof. We shall use the following equalities: f ( t ) = t , f ( t ) = t ; f ′ ( t ) = f ′ ( t ) = 3 . Let x ∈ S r ( t ) \ P , i.e., x = t + γ with | γ | p = r . We have | f ( x ) − t | p = | f ( x ) − f ( t ) | p = r · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γt + a γt + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p . (5.3)If r = h then using the property 3.1) we get from (5.3) that f ( x ) ∈ S r ( t ). In case r = h we use the p -adic version of the inverse function given by Theorem 2.2: assume that thereexists x ∈ S h ( t ) such that y = f ( x ) / ∈ S h ( t ), then this by the inverse function theoremgives the following contradiction: h = | x − t | p = | f − ( y ) − f − ( t ) | p = (cid:12)(cid:12)(cid:12)(cid:12) f ′ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) p | y − t | p = | y − t | p = h. (cid:3) Case p = 3 . In this case each fixed point t , t of g is an attractive point of g . Theorem 5.4. If p = 3 then (a) If r < h then for any x ∈ S r ( t ) \ P , lim n →∞ f n ( x ) = t and lim n →∞ f n +1 ( x ) = t ; for any x ∈ S r ( t ) \ P , lim n →∞ f n ( x ) = t and lim n →∞ f n +1 ( x ) = t . (b) If r > h then f ( S r ( t ) \ P ) ⊆ S r ( t ) , f ( S r ( t ) \ P ) ⊆ S r ( t ) . -ADIC (2 , (c) If r = h then for any x ∈ S h ( t ) \ P there exists ν = ν ( x ) ≥ h such that f n ( x ) ∈ S ν ( t ) and f n +1 ( x ) ∈ S ν ( t ) ; for any y ∈ S h ( t ) \ P there exists µ = µ ( y ) ≥ h such that f n ( y ) ∈ S µ ( t ) and f n +1 ( y ) ∈ S µ ( t ) .Proof. Let x ∈ S r ( t ) \ P , i.e., x = t + γ with | γ | = r . We have | f ( x ) − t | = | f ( x ) − f ( t ) | = r · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γt + a γt + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ϕ ( r ) = r, if r > h, ≥ r, if r = h, r h , if h < r < h, ≤ r , if r = h , r , if r < h . (5.4)For f ( x ) we have | f ( x ) − t | = | f ( x ) − f ( t ) | = | f ( x ) − t | · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − t t + a f ( x ) − t t + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ϕ ( ϕ ( x )) = ϕ ( r ) , if ϕ ( r ) > h, ≥ ϕ ( r ) , if ϕ ( r ) = h, ( ϕ ( r )) h , if h < ϕ ( r ) < h, ≤ ϕ ( r )3 , if ϕ ( r ) = h , ϕ ( r )3 , if ϕ ( r ) < h . Iterating this argument we obtain the following formulas for x ∈ S r ( t ) \ P : | f n ( x ) − t | = ϕ n ( r ) , | f n +1 ( x ) − t | = ϕ n +1 ( r ) . (5.5)Thus the dynamics of the radius r of the spheres is given by the function ϕ : [0 , + ∞ ) → [0 , + ∞ ), which is defined in formula (5.4). The following properties of ϕ are obvious:1 o . The set of fixed points of ϕ ( x ) is Fix( ϕ ) = { } ∪ ( h, + ∞ ) ∪ { h : if ϕ ( h ) = h } ;2 o . If ϕ n ( h ) = h , ∀ n = 1 , . . . , m , for some m ∈ N and ϕ m +1 ( h ) = h ∗ > h then ϕ k ( h ) = h ∗ for all k ≥ m + 1;3 o . The fixed point r = 0 is attractive with basin of attraction [0 , h ), independentlyon the value ϕ ( h ) ≤ h .Now using (5.5) it is easy to see that the assertion (a) follows from property 3 o ; theassertion (b) follows from property 1 o and (c) follows from 2 o . (cid:3) The case with two fixed points
In this section we consider the case c = 1, c = 0, d − acd + bc = 0 then the function f has two fixed points x and x (see (3.2)). We denote P = { x ∈ C p : ∃ n ∈ N ∪ { } , f n ( x ) = ˆ x } , for c = 1 . For any x ∈ C p , x = ˆ x , by simple calculations we get | f ( x ) − x i | p = | x − x i | p | c | p · | α ( x i ) + x − x i | p | β ( x i ) + x − x i | p , i = 1 , , (6.1)where α ( x ) = cx + 2 dx + ad − bccx + d , β ( x ) = cx + dc . Denote α i = | α ( x i ) | p , β i = | β ( x i ) | p , i = 1 , . Consider the following functions:For 0 ≤ α < β define the function ϕ α,β : [0 , + ∞ ) → [0 , + ∞ ) by ϕ α,β ( r ) = 1 | c | p αβ r, if r < α,α ∗ , if r = α, r β , if α < r < β,β ∗ , if r = β,r, if r > β, where α ∗ and β ∗ are some given numbers with α ∗ ≤ α β , β ∗ ≥ β .For 0 ≤ β < α define the function φ α,β : [0 , + ∞ ) → [0 , + ∞ ) by φ α,β ( r ) = 1 | c | p αβ r, if r < β,β ′ , if r = β,α, if β < r < α,α ′ , if r = α,r, if r > α, where α ′ and β ′ some positive numbers with α ′ ≤ α , β ′ ≥ α .For α ≥ ψ α : [0 , + ∞ ) → [0 , + ∞ ) by ψ α ( r ) = 1 | c | p ( r, if r = α, ˆ α, if r = α, where ˆ α is a given number.Using the formula (6.1) we easily get the following: Lemma 6.1. If x ∈ S r ( x i ) , then the following formula holds | f n ( x ) − x i | p = ϕ nα i ,β i ( r ) , if α i < β i ,φ nα i ,β i ( r ) , if α i > β i ,ψ nα i ( r ) , if α i = β i . n ≥ , i = 1 , . -ADIC (2 , Thus the p -adic dynamical system f n ( x ) , n ≥ , x ∈ C p , x = ˆ x is related to the realdynamical systems generated by ϕ α,β , φ α,β and ψ α . Now we are going to study these(real) dynamical systems. Lemma 6.2.
The dynamical system generated by ϕ α,β ( r ) , α < β has the following prop-erties:
1. Fix( ϕ α,β ) = { }∪ { r : r > β } ∪ { β ∗ : if β = β ∗ } , for | c | p = 1 , { r : r < α ; if α = | c | p β } ∪ { α : if α ∗ = | c | p α } ∪ {| c | p β : if α < | c | p β } , for | c | p < , { β : if β ∗ = | c | p β } , for | c | p > , ;2. For | c | p = 1 , independently on α , we have lim n →∞ ϕ nα,β ( r ) = , for all r < β,r, for all r > β,β ∗ , if r = β ;3. If | c | p > , then If β ∗ = | c | p β , then lim n →∞ ϕ nα,β ( r ) = 0 , for any r ≥ If β ∗ = | c | p β , then lim n →∞ ϕ nα,β ( r ) = ( , for any r / ∈ B = {| c | kp β : k = 0 , , , . . . } ,β, if r ∈ B ;4. If | c | p < , then If α < | c | p β , then lim n →∞ ϕ nα,β ( r ) = , for all r < | c | p β,r, for r = | c | p β, + ∞ , if r > | c | p β ;4.b) If α = | c | p β , α ∗ = | c | p α , then lim n →∞ ϕ nα,β ( r ) = (cid:26) r, for all r ≤ α, + ∞ , if r > α ;4.c) If α = | c | p β , α ∗ = | c | p α , then lim n →∞ ϕ nα,β ( r ) = r, for all r < α,α ∗ / | c | p , for r = α, + ∞ , if r > α ; If α > | c | p β , α ∗ = | c | p α , then lim n →∞ ϕ nα,β ( r ) = , for r = 0 ,α, for r ∈ L = { ( | c | p β ) k α − k , k ≥ } , + ∞ , if r / ∈ L. If α > | c | p β , α ∗ > | c | p α , then lim n →∞ ϕ nα,β ( r ) = (cid:26) , for r = 0 , + ∞ , if r > . If α > | c | p β , α ∗ < | c | p α , then there exists k ≥ such that the sequence C = { α, ϕ α,β ( α ) , . . . , ϕ k − α,β ( α ) } is a k -cycle of ϕ α,β and lim n →∞ ϕ nα,β ( r ) = , for r = 0 , ∈ C , for r ∈ U = { r : ∃ n ∈ N , ϕ nα,β ( r ) ∈ C} , + ∞ , if r / ∈ U. Proof.
1. This is the result of a simple analysis of the equation ϕ α,β ( r ) = r .Proofs of parts 2-4 follow from the property that ϕ α,β ( r ), r = α, β is an increasingfunction. (cid:3) Lemma 6.3.
The dynamical system generated by φ α,β ( r ) , α > β has the following prop-erties: A. Fix( φ α,β ) = { }∪ { r : r > α } ∪ { α : if α = α ′ } , for | c | p = 1 , { α : if α ′ = | c | p α } , for | c | p < , { α/ | c | p } , if α > | c | p β, | c | p > , { β : if β ′ = | c | p β } , for α < | c | p β, | c | p > . ;B. For | c | p = 1 , we have B.a) If α = α ′ , then lim n →∞ φ nα,β ( r ) = , for r = 0 ,α, for all r ≤ α,r, for all r > α, ;B.b) If α = α ′ , then there exists k ≥ such that the sequence α, α ′ = φ α,β ( α ) , φ α,β ( α ) , . . . , φ k − α,β ( α ) is a k -cycle for φ α,β ( r ) and lim n →∞ φ nα,β ( r ) converges to the k -cycle for any r ≤ α . Moreover the limit is equal to r for any r > α . -ADIC (2 , C. If | c | p < , then C.a) If α ′ = | c | p α , then lim n →∞ φ nα,β ( r ) = , if r = 0 ,α, for any r ≤ α, + ∞ , for any r > α ;C.b) If α ′ = | c | p α , then there exists k ≥ such that the sequence P = { α, α ′ = φ α,β ( α ) , φ α,β ( α ) , . . . , φ k − α,β ( α ) } is a k -cycle for φ α,β ( r ) and lim n →∞ φ nα,β ( r ) = , if r = 0 , ∈ P , if r ∈ W = { r : ∃ n ∈ N , φ nα,β ( r ) ∈ P } , + ∞ , if r / ∈ W ;D. If | c | p > , then D.a) If α > | c | p β , then lim n →∞ φ nα,β ( r ) = (cid:26) α | c | p , for all r > , , for r = 0 , ;D.b) If α < | c | p β , β ′ = | c | p β , then lim n →∞ φ nα,β ( r ) = ( β, for all r ∈ M = {| c | kp β, k ≥ } , , for r / ∈ M, ;D.c) If α < | c | p β , β ′ = | c | p β , then lim n →∞ φ nα,β ( r ) = 0 , for r ≥ . Proof.
Since φ α,β ( r ) is a piecewise linear function the proof consists of simple computa-tions, using the graph of the function and varying the parameters α, β, | c | p . (cid:3) The following lemma is obvious:
Lemma 6.4.
The dynamical system generated by ψ α ( r ) , α ≥ has the following properties: (i) Fix( ψ α ) = { } ∪ (cid:26) { r : r = α } ∪ { α : if α = ˆ α } , for | c | p = 1 , { α : if ˆ α = | c | p α } , for | c | p = 1 . ;(ii) If | c | p = 1 , then lim n →∞ ψ nα ( r ) = r, for any r = α, α = ˆ α, ˆ α, for r = α, α = ˆ α,r, for any r ≥ , α = ˆ α, (iii) If | c | p > , then lim n →∞ ψ nα ( r ) = , for any r ≥ , | c | p α = ˆ α,α, for r ∈ H = {| c | kp α : k ≥ } , | c | p α = ˆ α, , for r / ∈ H, | c | p α = ˆ α, (iv) If | c | p < , then lim n →∞ ψ nα ( r ) = , for r = 0 , + ∞ , for any r > , | c | p α = ˆ α,α, for r ∈ H = {| c | kp α : k ≥ } , | c | p α = ˆ α, + ∞ , for r / ∈ H, | c | p α = ˆ α. Now we shall apply these lemmas to the study of the p -adic dynamical system generatedby f .For x ∈ S α i ( x i ), we denote α ∗ i ( x ) = α i · | α ( x i ) + x − x i | p | β ( x i ) + x − x i | p , i = 1 , . For x ∈ S β i ( x i ), we denote β ∗ i ( x ) = β i · | α ( x i ) + x − x i | p | β ( x i ) + x − x i | p , i = 1 , . Using Lemma 6.1 and Lemma 6.2 we obtain the following
Theorem 6.5. If α i < β i and x ∈ S r ( x i ) , i = 1 , , then the p -adic dynamical systemgenerated by f has the following properties: The following spheres are invariant with respect to f : S r ( x i ) , if r > β i , | c | p = 1 ,S r ( x i ) , if r < α i , α i = | c | p β i ; | c | p < ,S | c | p β i ( x i ) , if α i < | c | p β i , | c | p < . ;2. For | c | p = 1 , we have lim n →∞ f n ( x ) = x i , for all r < β i ,f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for all r > β i , lim n →∞ f n ( x ) ∈ S β ∗ i ( x ) ( x i ) , if r = β i ;3. If | c | p > , then If β ∗ i ( x ) = | c | p β i , then lim n →∞ f n ( x ) = x i , for any r ≥ -ADIC (2 , If β ∗ i ( x ) = | c | p β i , then lim n →∞ f n ( x ) = x i , for any r / ∈ B, ∈ S β ∗ i ( x ) | c | p ( x i ) , if r ∈ B ;4. If | c | p < , then If α i < | c | p β i , then lim n →∞ f n ( x ) = x i , for all r < | c | p β i ,f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for r = | c | p β i , lim n →∞ | f n ( x ) − x i | p = + ∞ , if r > | c | p β i ;4.b) If α i = | c | p β i , α ∗ i ( x ) = | c | p α i , then f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for all r ≤ α i , lim n →∞ | f n ( x ) − x i | p = + ∞ , if r > α i ;4.c) If α i = | c | p β i , α ∗ i ( x ) = | c | p α i , then f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for all r < α i , lim n →∞ f n ( x ) ∈ S α ∗ i ( x ) / | c | p ( x i ) , for r = α i , lim n →∞ | f n ( x ) − x i | p = + ∞ , if r > α i ;4.d) If α i > | c | p β i , α ∗ i = | c | p α i , then lim n →∞ f n ( x ) ∈ S α ∗ i ( x ) / | c | p ( x i ) , for r ∈ L = { ( | c | p β i ) k α − ki , k ≥ } , lim n →∞ | f n ( x ) − x i | p = + ∞ , if r / ∈ L. If α i > | c | p β i , α ∗ i ( x ) > | c | p α i , then lim n →∞ | f n ( x ) − x i | p = + ∞ , if r > . If α i > | c | p β i , α ∗ i < | c | p α i , then there exists k ≥ such that the limitingtrajectory of f n ( x ) , n ≥ will periodically visit the spheres S ϕ jαi,βi ( α i ) ( x i ) , j = 0 , , . . . , k − in the following way: if r ∈ U = { r : ∃ n ∈ N , ϕ nα i ,β i ( r ) ∈ C} ,then S α i ( x i ) → S ϕ αi,βi ( α i ) ( x i ) → · · · → S ϕ k − αi,βi ( α i ) ( x i ) → S α i ( x i ) , and lim n →∞ | f n ( x ) − x i | p = + ∞ , if r / ∈ U. By Lemma 6.1 and Lemma 6.3 we obtain the following
Theorem 6.6. If α i > β i and x ∈ S r ( x i ) , i = 1 , , then the p -adic dynamical systemgenerated by f has the following properties: A. The following spheres are invariant: S r ( x i ) , if r > α i , | c | p = 1 ,S α/ | c | p ( x i ) , if α i > | c | p β i , | c | p > . B. For | c | p = 1 , we have B.a) If α i = α ∗ i ( x ) , then lim n →∞ f n ( x ) ∈ S α ∗ i ( x ) ( x i ) , for all r ≤ α i ,f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for all r > α i ;B.b) If α i = α ∗ i ( x ) , then there exists k ≥ such that the limiting trajectory of f n ( x ) , n ≥ will periodically visit the spheres S φ jαi,βi ( α i ) ( x i ) , j = 0 , , . . . , k − in the following way: S α i ( x i ) → S φ αi,βi ( α i ) ( x i ) → · · · → S φ k − αi,βi ( α i ) ( x i ) → S α i ( x i ) , f or any r ≤ α i and lim n →∞ f n ( x ) ∈ S r ( x ) , f or any r > α i . C. If | c | p < , then C.a) If α ∗ i ( x ) = | c | p α i , then f ( S r ( x i ) \ P ) ⊂ S α ( x i ) , for any r ≤ α i , lim n →∞ | f n ( x ) − x i | p = + ∞ , for any r > α i ;C.b) If α ∗ i ( x ) = | c | p α i , then there exists k ≥ such that the limiting trajectory of f n ( x ) , n ≥ will periodically visit the spheres S φ jαi,βi ( α i ) ( x i ) , j = 0 , , . . . , k − in the following way: if r ∈ W = { r : ∃ n ∈ N , φ nα i ,β i ( r ) ∈ P } , then S α i ( x i ) → S φ αi,βi ( α i ) ( x i ) → · · · → S φ k − αi,βi ( α i ) ( x i ) → S α i ( x i ) , lim n →∞ | f n ( x ) − x i | p = + ∞ , if r / ∈ W ;D. If | c | p > , then D.a) If α i > | c | p β i , then lim n →∞ f n ( x ) ∈ S α | c | p ( x ) , for all r > , D.b) If α i < | c | p β i , β ∗ i ( x ) = | c | p β i , then lim n →∞ f n ( x ) ∈ S β i ( x i ) , for all r ∈ M = {| c | kp β i , k ≥ } , lim n →∞ f n ( x ) = x i , for r / ∈ M ;D.c) If α i < | c | p β i , β ∗ i ( x ) = | c | p β i , then lim n →∞ f n ( x ) = x i , for r ≥ . By Lemma 6.1 and Lemma 6.4 we get -ADIC (2 , Theorem 6.7. If α i = β i , and x ∈ S r ( x i ) , i = 1 , , then the dynamical system generatedby f has the following properties: (i) For any r = α i the sphere S r ( x i ) is an invariant set. (ii) If | c | p = 1 , then (ii.a) If α i = α ∗ i ( x ) , then f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for all r = α i ; f ( S r ( x i ) \ P ) ⊂ S α ∗ i ( x ) ( x i ) , if r = α i ;(ii.b) If α i = α ∗ i ( x ) , then f ( S r ( x i ) \ P ) ⊂ S r ( x i ) , for all r = 0;(iii) If | c | p > , then (iii.a) If α ∗ i ( x ) = | c | p α i , then lim n →∞ f n ( x ) = x i , for any r ≥ If α ∗ i ( x ) = | c | p α i , then lim n →∞ f n ( x ) = x i , for any r ∈ H = {| c | kp α i : k ≥ } , lim n →∞ f n ( x ) ∈ S α ( x i ) for r / ∈ H. (iv) If | c | p < , then (iv.a) If α ∗ i ( x ) = | c | p α i , then lim n →∞ | f n ( x ) − x i | p = + ∞ , for any r > , (iv.b) If α ∗ i ( x ) = | c | p α i , then if r ∈ H = {| c | kp α i : k ≥ } , then lim n →∞ f n ( x ) ∈ S α i ( x i ) , for any r ∈ H, lim n →∞ | f n ( x ) − x i | p = + ∞ , for any r / ∈ H. Acknowledgments
U.Rozikov thanks the universit´e du Sud Toulon Var for supporting his visit to Toulonand the Centre de Physique Th´eorique de Marseille for kind hospitality. He also would liketo acknowledge the hospitality of the ”Institut f¨ur Angewandte Mathematik”, Universit¨atBonn (Germany). This work is supported in part by the DFG AL 214/36-1 project(Germany).
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S. Albeverio, Institut f¨ur Angewandte Mathematik and HCM, Universit¨at Bonn, En-denicher Allee 60, 53115 Bonn, Germany; CERFIM (Locarno).
E-mail address : [email protected] U. A. Rozikov and I.A. Sattarov, Institute of mathematics and information technologies,29, Do’rmon Yo’li str., 100125, Tashkent, Uzbekistan.
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