Cycles of monomial and perturbated monomial p-adic dynamical systems

Discrete dynamical systems over the field of p-adic numbers are considered. We will concentrate on the study of periodic points of monomial and perturbated monomial system. Similarities between these two kinds of systems will be investigated. The conditions of the perturbation and the choice of the prime number p plays an important role here. Our considerations will lead to formulas for the number cycles of a specific length and for the total number of cycles. ‘Ve will also study the distribution of cycles in the different p-adic fields.


Introduction
Discrete dynamical systems have a lot of applications, for example in biology and physics, [7]. Dynamical systems over the p-adic numbers (see for example [11] and [2] for a general introduction to p-adic analysis) can also be used for modelling psychological and sociological phenomena, see [7] and [6]. Especially, in [6] a modell of the human memory, using p-adic dynamical system, is presented.
The most studied p-adic dynamical systems are the so called monomial systems. A (discrete) monomial system is defined by a function f (x) = x'~. In [9] there is a stochastic approach to such systems. In [10] dynamical systems (not only monomial) over finite field extensions of the p-adic numbers are considered.
By using theorems from number theory, we will be able to prove formulas for the number of cycles of a specific length to a given system and the total number of cycles for monomial dynamical systems. We will also investigate the number of cycles of a specific length to a system for different choices of the prime number p. Here some remarkable asymptotical results occur.
We will also study perturbated monomial dynamical systems defined by functions, fq(x) = xn + q(x), where the perturbation q(x) is a polynomial whose coefficients have small p-adic absolute value. We investigate the connection between monomial and perturbated monomial systems. In this investigations we will use Hensel's lemma. As in the monomial case the interesting dynamic of some perturbated systems are located on the unit sphere in Qp. Sufficient conditions on the perturbation for the two systems to have similar properties are derived. By similar properties we mean that there is a one to one correspondence between fixed points and cycles of the two kinds of systems.

Properties of monomial systems
Everywhere below we will use the following notations: The field of p-adic numbers is denoted by Qp, the ring of p-adic integers is denoted Zp. We will use ~ ~ ~p to denote the p-adic valuation. The sphere, ball and open ball with radius p and center at a, with respect to the p-adic metric induced by the padic valuation, are denoted by Sp(a), Up(a) and respectively.
We use the notation a = b (mod pkZp) if and only if b|p 1/pk. In this article we will first consider the dynamical systems f : Qp where f(x) = x'~ (2.1) and n E N such that ?~ ~ 2. In [7] there is an extensive investigation of these systems. Most of the theorems in this section come from this book. In the following we will use the notation fr to denote the composition of f with itself r times. By limr~~ fr(x) = Xo we mean that limr~~|fr(x) -x0|p = 0.
Definition 2.1. Let zr = If Xr = XQ for some positive integer r then 2*0 is said to be a periodic point of f If r is the least natural number with this property, we call r the period of xo. A periodic point of period 1 is called a fixed point of f. . Definition 2.2. Let r be a positive integer. The set, = {xe, ... , xr-1} of periodic points of period r is said to be a cycle to the dynamical system determined by f if xo = f (xr_1) and xj = for 1 j r -l. The length of the cycle is the number of elements in ~y. Definition 2.3. A fixed point xo to a function f is said to be an attractor if there exists a neighbourhood V(xo) such that for every y E V (xo) holds that lim x0.
By the basis of attraction we mean the set = {y E x0~Ĩ t is known that for a monomial system (2.1) 0 and oo are attractors and that .4(0) = U1(0) and A(oo) = Qp B Ul(0). The rest of the periodic points are located on SI (0). Fixed points of (2.1) on 5'~ (0) are solutions of the equation =  It follows directly from the definition of the periodic points that the set of solutions to equation (2.2) not only contains the periodic points of period r but also the periodic points with periods that divides r. We have the following theorem about the roots of (2.2) in Qp. (We use gcd(m, n) to denote the greatest common divisor of two positive integers m and n. ) Theorem 2.4. The equation x~ =1 has gcd(k, p-1 ) solutions in Qp when p > 2. If p = 2 then xk = 1 has two solutions (x = 1 and x = -1) if k is even and one solution ~x =1) if k is odd. Let N(n, r, p) denote the number of periodic points of period r in (2.1).
Each cycle of length r contains r periodic points with period r. If we denote by K(n, r, p) the number of cycles of length r then K(n~ r~ p) = N(n , r, p) /r. (2~3) In [7] we find the following theorem about the existens of cycles. Theorem 2.5. Let p > 2 and let mj = gcd(n~ -1,p -1). . The dynamical system f(x) = xn has r-cycles (r > 2) in Qp if and only if mr does not divide any m~, r 2014 1. . We have the following relation between mj, N(n, j, p) and K(n, j, p) mj = N(n, i, p) = 2K(n, i, P). (2.4) ilJ zh Here follow some more facts about the monomial systems: Theorem 2.6. If p = 2 then the dynamical system (2.1) has no cycles of order r > 2.
Proof. If n is even then it follows from Theorem 2.4 that (2.1) has only one fixed point in Q2. It also follows that nr is even for all r 2 and this implies that fr(x) = xnr only has one fixed point in Qz which also is the fixed point of f ( x) = xn. Hence f has no periodic points of period r. The case when n is odd is proved in a similar way. We will now determine the maximum of numbers of cycles, of any length, in Qp for a fixed p. Let n > 2 be a natural number. Denote by p* (n) the number we get if we remove, from the prime factorisation of p -~ 1, the factors dividing n. That is p* (n) is the largest divisor of p -1 which is relatively prime to n. We also recall the definition of Euler's p-function and Euler's Theorem.
Definition 3.5. Let n be a positive integer. Henceforth, we will denote by cp(n) the number of natural numbers less than n which are relatively prime to n. The function cp is called Euler's p-function.
An equivalent definition of p is that cp(n) is the number of elements in 1Fn which are not divisors of zero. If p is a prime number then 03C6(pl) = -1). . Lemma 3.7. With the above notation we have for each r E N gcd(nrl,p -1) = gcd(nr -1,p*(n)). (3.5) Proof. Since nr -1--1 (mod q) if q ~ n it follows that we can remove the prime factors from p -1 which divides n and it would not change the value of gcd(nT -l,p -1). D Lemma 3.8. There is a least integer (n), such that = p*(n). Proof. Since gcd(n,p*(n)) = 1 it follows from Theorem 3.6 that n03C6(p*(n)) -1 (modp*(n)). It is then clear that there exists a smallest such that 1 (modp*(n)) and (n) 03C6(p*(n)). (It is also true that ) /?(?* (~)).) Hence p*(n) ( -1 and the theorem follows. ll Theorem 3.9. Let p > 2 be a fized prime number, let n > 2 be a natural Proof. We first prove that N(n, r, p) = 0 if r > r(n). Since I, p -1) = p*(n) and every mr = p*(n), r > it follows from Theorem 2.5 that N(n, r, p) = 0. Next we want to prove that if r f f(n) then N(n, r, p) = 0. Let dl be a divisor of p* (n). Let q be the least integer such that n~ -1 = 0 (mod Ll ) . Since = 1 (modp*(n)) it follows that 1 (mod l1). By the division algorithm we have f(n) = kq + rl. This implies that 1 == nkq+rl -(nq)knr1 -nrl (mod l1).
Since q was the least non-negative integer such that 1 (mod l1) it follows that rl = 0. That is q ~ r(n).
The only possible values of are the divisors of p* (n). In the paragraph above we showed that the least number q such that gcd(nql,p 2014 1) = li, where l1 | p*(n), must be a divisor of r(n). Hence if r f r(n) then N(n, r, p) = 0.
So far we have proved that R N(n, r, p) = ~ N(n, r, p). r=1 r|(n) We have left to prove that L N(n,r,p) = p*(n). °r lr(n) From (2.4) we know that gcd(nr -1,p*(n)) = dlr If we set r = f(n) we have proved the theorem. D Corollary 3.10. Let p > 2. The dynamical system (2.1) has p* (n) periodic points in Qp. . Theorem 3.11. Let p > 2. The total number,K Tot(n,p), of cycles of (2.1) is given by rjr r~r r d~r Proof. . From the proof of Theorem 3.9 we know that we only have cycles of lengths that divides f(n). From (3.4) it follows that K(n,r,p) = 1 r 0 3 A 3( d ) g c d ( n r / d -1 , p -1 ) . d~r The theorem is proved. D Example 3.12. Let us consider the monomial system f(x) = x2 (n = 2).
If p = 137 then it follows from Corollary 3.10 that the dynamical system has p* (2) = 17 periodic points and from Theorem 3.11 it follows that it has KTot (2,137) = 3 cycles. In fact the monomial system f(x) = ~2 has one cycle of length 1 (one fixed point) and two cycles of length 8.
If we consider the same system, but let p = 1999 instead, then the number of periodic points is p*(2) = 999 and the number of cycles is KTot {2,1999) = 31. In fact the system has one cycle of length 1, 2, 6 and 18 and also 27 cycles of length 36. Example 3.13. Let us now instead consider the dynamical system f(x) = x3. If p =137 then there are 136 periodic points and 13 cycles. In fact we have two fixed points, three cycles of length 2 and 8 cycles of length 16. If instead p = 1999 then there are two fixed points and four cycles of length 18, so we have 74 periodic points and six cycles.  We will prove that the asymptotical inclination is in fact a constant and that this constant can be expressed rather easily. We will prove the following theorem:  Theorem 4.2. Let n and r be positive integers such that n > 2 and r > 2.
We then have where is Möbius function.
Before we start to prove this theorem we need some results from number theory. We will use the arithmetical functions cp, ,u and r.
We first recall some simple connections between cp and which will be useful to us later on, see [1]. . (4.2) dJTi We will also need some results from number theory concerning the distribution of primes. Henceforth we will use the notation f ( Definition 4.4. For x > 0 we define 7r(T) to be the number of primes less or equal to x. Theorem 4.5 (Prime number theorem). Let be as above then x log x (4.3) The proof of this theorem can be found in [4], [13] and [1]. . Definition 4.6. Let k and a be positive integers such that gcd(a, k) = 1.
Let be the number of primes p less or equal to x such that p = a (mod k).
The number is the number of primes less or equal to x which can be written as kn + a, where n is a natural number. Dirichlet proved the following theorem: Theorem 4.7. If gcd(a, k) =1 then there are infinitely many prime numbers p which can be written p = kn + a. .
The following theorem is a generalization of the prime number theorem.

.4)
A proof of this theorem can be found in [14].
We are now ready to prove the main part of Theorem 4.2. We state it as a theorem.    dlr For the distribution of cycles we have the following theorem, which follows directly from Theorem 4.2.
Theorem 4.10. We have l i m 1 M K ( n , r , p ) = 1 r (d)(n(r/d)-1). (4'io) 5 Perturbation of monomial systems In [7] there is an extensive investigation of monomial dynamical systems over the field of p-adic numbers, Qp. In this section we will follow the ideas from [7] for investigations of perturbations of such systems. We will use the following theorems a lot.   Since each term in the second factor on the right in the equation above contains at least one x or one a we have for all x E Qp, such that |x|p 1, that there exists a real number c 1 such that 03B1|p 03B1|p.

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In the above theorem we only need q to be a perturbation, not a kperturbation.
Theorem 5.6. Consider the dynamical system (5.5) and assume that the degree of q is less or equal to n. We have that Since the parenthesis in the last expression is positive, there is a constant, c > 1, such that > c|x|p. for all x satisfying |x|p > 1. By induction it is easy to prove that |frq(x)|p > cr|x|p.
Hence, ~ 00 as r ~ 00 if |x|p 1. If 1 it follows directly from the strong triangle inequality that 1 and by induction that 1.
D If we assume that the degree of the perturbation polynomial q is less or equal to n it follows from Theorem 5.4 and Theorem 5.6 that A(a) = Ui (0). If we assume that deg q fi n, we can say that a and oo are attractors to the dynamical system f(x) = xn + q(x), and that the basins of attraction are and Qp B U1 (o) respectively. If deg q > n we do not always have A(a) = Ui (0), see the following example. Example 5.7. Let p be a fi.xed prime number and let n > 2 be an integer such that p f n -1. Let q(x) = cxn+1, where c = 03A3~i=1 (p -1 ) p$. It is clear that q is a perturbation to the dynamical system f(x) = xn. Consider the dynamical system fq(x) = xn + q(x). Let x =1 /p = p) then 00 fq(1/p) = 1/pn + 0 3 A 3 ( p -1 ) p i -( n + 1 ) = 0. i=l From Theorem 5.5 it follows that there is a fixed point a E Ui (0) and that Ui (0) C A(a). Since 0 E A(a) it follows that 1/p E A(a).
We will now start to investigate the behaviour of the dynamical system on the sphere S'1(0). . Theorem 5.8. Let a E Sl(O) be a fixed point of the dynamical system (2.1). Then there exists a fixed point, a E Zp, to the dynamical system (5.5) such that a == a (mod pk+1Zp).
Proof. Let 03C8(x) = f(x)-x = xn-x and let 03C6(x) = fq(x)-x = Since it follows that 03C6(a) = 03C8(a) + q(a) = 03C8(a) and since 03C8(a) = 0 (a is a fixed point to that dynamical system) it follows that 0 (mod p2k+1Zp). We also have that cp'(x) = 03C8'(x) = nxn-1 -  (that is, a fixed point to f {x)) and a ~ a (mod pk+1Zp). D We now prove the converse to this theorem. Theorem 5.9. If a E Sl (0) is a fixed point to the dynamical system (5.5), then there exists a fixed point, a, (a root of unity) to {2.1) such that a ã (mod pk+1Zp).
Proof. yVe will use Theorem 5.I to prove this. Let Proof. Let a and b (a ~ b) be two fixed points in S~(0) to the monomial dynamical system (2.1). According to Theorem 5.8 there are fixed points a and 03B2 on S1(0) to (5.5) such that la -03B1|p 1/p and |b -03B2|p 1/p. From Theorem 2.7 it follows that la -b~~ = 1. We therefore havẽ (a-a)+(a'b)+(b'~)~r=1~ > since j(aa) + (6 -/3)~p ~ 1/p. Hence /3. The second part of the theorem is proved similarily. Q Remark 5.11. If ~q~ > 1, Theorem 5.10 no longer holds. Example 5.12. Let p = 3, f(x) = x2 and fq(x) = x2 -2. The dynamical system f has only one fixed point (x = 1) on Sl (0). But the dynamical system f q has the fixed points x = 2 and x = -I on Sl (0) . 6 Cycles of perturbated systems In this section we will start to study the dynamical system = x'~ + q(x), (6.I) where q is a perturbation, f(x) = = To study cycles of length r to this system, we look for fixed points to f q . We can write where qr is a new perturbation. Let Cr denote the set of fixed points to (6.2). All periodic points of period r are contained in Cr, but this set also contains periodic points of periods that divides r. 9q,r(x) = xnr -x + qr(x) = gr(x) + qr(x).
First, let us assume that a is a fixed point to f a that is gr(a) = 0. The fact that = 1 implies that and that g'r(a) = = nr -1. .
The theorem now follows from Hensel's lemma. 0 We have not this results in the case nr -1 ~ 0 (modp). . Example 6.2. Let p = 3, f(x) = x2 and fq(x) = x2 -39/4. We are going to show that f has no cycles of length 2 but f q has one cycle of length 2.
From the fact that gcd(n2 -1,p -1) = 1 we immediately have that f has no cycles of length 2. Since > has two fixed points, x = 5/2 and x = -7/2, and none of them are fixed points to fq(x) it follows that fq has one cycle of length 2. Theorem 6.3. Let p be a fixed prime number and let n E N and n > 2. If p f n there is a least r such that nr -1-0 (mod p) and 2 r p -l. . If r f r then nr -1 ~ 0 (modp).
Proof. . Consider the multiplicativ group I~ _ ~ I, Z, ... , p-1 }, of the field of residue classes Fp. We know that IF~ is a cyclic group under multiplication. Let d be the remainder when n is divided by p, of course d E F*p . Due to Fermat we have cP~"~ 2014 1 = 0 (mod p). That is, there exits r such that nr-1= 0 (modp). Since the set {2,3,... ,p-1~ is finite there exists a least r, say r. It is clear that r is the order of the cyclic subgroup generated by d. (Accordning to the Theorem of Lagranges r must be a divisior of p -1. ) Assume that nr -1 ~ 0 (mod p), then there is a cyclic subgroup of order r.
Let us now study two cases: (i) If r = 21 then 2r -1 = 0(mod3), so gcd(2r-1, 3) = 3. (ii) If r = 2l+1 then 2''-1 = 1 (mod 3), so gcd(2r-1, 3) _ 1. We can now make the following conclusions: The dynamical system f q has cycles of order 2 and there exists no cycles of order r if 2 f r and 3 f r (or r ~ 3 (mod6)). . Example 6.6. Let p = 43 and let f and f q be as above. One can show (or use a computer) that 2r --1 ~ 0 (mod 43) if and only if 14 f r. We have the following values for mr: The dynamical system f q therefore has cycles of order 2, 3 and 6. If r > 3, r ~ 6 and 14 f r, then the dynamical system f q has no cycles of order r.
To get more information about the cycles of the dynamical system (6.1) we have to use stronger conditions on the perturbation polynomial. gq,r(x) = fQ (x) " x = gr(x) + qr(x).
Let us first assume that a E Sl (0) is a fixed point to that is gr (a) = 0.
Since (d/dx)gq,r(x) = nrxnr-1 -1 + q; (x) we also have (d/dx)gq,r(a) = . 0 (mod p03BAZp) and (d/dx)gq,r(a) ~ 0 (mod p03BA+1Zp). According to Theorem 5.1 there is a E S1(o) such that gq,r(a) = 0 and ja -03B1|p 1/p03BA+1. Assume now that b E Si(0) is a fixed point to f;. . If we observe that gr (x) = gq,r (x) -qr(x), we can make the conclusion that gr(b) = -qr(b) ~ (mod p203BA+1Zp). The conditions in Theorem 5.1 are satisfied, so there exists /3 E S1(0) such that gr(03B2) = 0 and The proof is complete. D Observe that for p > 2 we have a one-to-one correspondence between the cycles of a specific length to the dynamical system f q and f. . This follows directly from Theorem 5.10. Before we present some examples we state some theorems which will help us in the construction of these examples. Theorem 6.8. Let p be a fixed prime number and let d > 2, n > 2. If p is not a divisor of n, then there is a least integer r such that nr -1 0 (mod pl), 1 r 03C6(pl) .
Assume nr = 1 (modp'). If we divide r by r we get r = cr + d. Since d f and r is the least integer such that nr ~ 1 (mod pl) we must have d = 0 and hence fir. The theorem is proven. D Theorem 6.9. Let n, m and I be positive integers. 11n -1 (mod m~ ), then nm == 1 (mod ml+1) . 1fm = p is a prime number and n ~ I (mod pl+1) then p is the least m such that nm =1 (mod pl+1). Proof. Since n = 1 (mod m') we can write n = qml + 1. If we assume that q is a 1-perturbation, that is ~133, then it follows from (6.3), (6.4) and Theorem 6.7 that if 6 f r then f q has no cycles of order r.
We can continue this investigations by repeating the above arguments. If we assume that q is a 2-perturbation then we can make the conclusion that f q has no cycles of length r if 18 f r.
More general, if we assume that ~q~ 1/3203BA+1 then there are no cycles of length r if 2 . 3" f r, by Theorem 6.10. Example 6.12. Let p = 7 and let fq(x) = x2 + q(x), where q is a perturbation. According to Theorem 6.3 the dynamical system f(x) = x2 has cycles only of length 2. Accordning to Example 6.5, f q has a cycles of length 2 and we also know that f q has no cycles of order r if r > 2 and 3 f r. Let us now assume that r = 3r1, where ri E ?L+, then 2~i -1 == 0(mod7). Since 8 ~ 1 (mod 49) we have that 7 is the least positive integer d such that 8d -1 (mod 49), by Theorem 6.9. We therefore have that 23(7r2+03B1) -1 ~ 0 (mod 49) if 1 03B1 6. If 1/p3 it follows from Theorem 6.7 that there are no cycles of order r to the dynamical system fq if r > 2 and 21 ~ r.
If we assume that 1/72"+1 then it follows from Theorem 6.7 and 6.10 that the dynamical system f q has no cycles of length r if 3 . 7" f r. Example 6.13. Let n = 10 and let p = 3. Since mr = = gcd(lOr -1,2) = 1 it follows from Theorem 2.5 that the dynamical system f(x) = xl° has no cycles. We have that nr -1 ~ 0 (mod 9) for every r > 2 and if 3 f r we have r~r -1 ~ 0 (mod 27). If we assume that 1/35 we have by Theorem 6.7 that fq has no cycles of length r if 3 ~ r. If 1/3~"+~ then fq has no cycles of length r if 3"-1 ~' r.
Example 6.14. Let n = 2 and p = 251. Computer calculations show that r = 50 is the least positive integer such that nr --1 -0 (mod 251). According to Theorem 2.5 we have that f only has cycles of lengths 4,20 and 100. Due to Theorem 6.1 we can make the conclusion that f q has cycles of order 4 and 20, and that f q has no cycles of order r if r 7~ 4, r ~ 20 and 50 f r. By using a computer we get that n~°° --1 y 0 (mod 2512). So, if q is a 1-perturbation we have according to Theorem 6.7 that f q also has a cycle of order 100.
It is possible to generalize the theorems in Section 3 and 4 to some perturbated monomial systems. Assume that p > 2. Let r denote the length of the longest cycle of f and let Nq(n, r, p) denote the number of periodic points on 81 (0) of period r of f q (a corresponding perturbated system). If nrj -1 = p03BAnj, p nj for all rj | then it follows from Theorem 5.10 that = if q is a x-perturbation.