FINITE DYNAMICAL SYSTEMS: A PROBABILISTIC APPROACH
ALEXANDER BECKWITH AND JOSEPH DICKENS
A DVISOR : C LAYTON P ETSCHE
A BSTRACT.
Since the behavior of large finite dynamical systems is difficult to observe and characterize, we
approach these number theoretic objects from a probabilistic point of view. By doing so, we develop
expectations about large random finite dynamical systems. In this paper we find and prove several
explicit and asymptotic formulas describing the growth of the set of periodic elements as set size
becomes arbitrarily large. We conclude this paper with several additional conjectures concerning
the asymptotic behavior of the set of periodic elements when we have a d-to-one function.
1. I NTRODUCTION
A finite dynamical system can be thought of as an iteration of a function on a set of elements.
More specifically, let φ : X → X be a function from a nonempty finite set X to itself. Then we call
the pair (X, φ) a finite dynamical system. Small dynamical systems can be represented visually by
a digraph like the one found in figure 1. Though a system’s characteristics can be easy to observe
for small set size, the behavior of large finite dynamical systems can be much more difficult to
describe. In the setting of very large finite dynamical systems, probability theory techniques can
be employed to describe these systems’ behavior. Calculations of mean and variance shed light on
the how a “typical” system behaves as its size become arbitrarily large.
Finite fields provide a number theoretic context for our calculations in the probabilistic setting.
We show that certain relationships between a set and functions on that set can produce predictable
behavior. With an understanding of the number theoretic behavior underlying finite dynamical
systems, we are able to computationally observe predictions made from probability theory calculations.
This paper begins with a brief introduction to finite dynamical systems. We then use probability
theory in the next section to calculate the mean and variance of expected number of periodic points.
We provide further support for our previous calculations using computer simulation. We end our
paper with several conjectures concerning probability and number theoretic aspects of our project.
We hope that these basic calculations provide a basis for further study of this intersection of number
and probability theory.
Date: 08/16/2013.
This work was done during the Summer 2013 REU program in Mathematics at Oregon State University.
1
2
Beckwith and Dickens
2
0
4
7
5
9
8
10
3
1
6
F IGURE 1. Digraph of (F11 , x2 ).
1.1. Definitions with basic example to illustrate.
Before proceeding, we will briefly provide several definitions that will guide our study of finite dynamical systems. Define a dynamical system (X, φ) as above. We use the term iteration
to refer to repeated composition of the function φ with itself: let α ∈ X and for n ∈ N denote
the n-fold composition φn (α) = (φ ◦ φ ◦ · · · ◦ φ)(α). After iterating φ across all elements of X,
X is partitioned into two sets of points. The first is the periodic set of X under φ, defined as
Per(X, φ) = {α ∈ X : α is periodic}. Every element of X in Per(X, φ) is in a cycle. Formally, α is
a periodic point of φ if φn (α) = α for some n ∈ N. The second set of (X, φ) is the set of strictly
preperiodic elements or tail elements. We denote the set of tail elements by Tail(X, φ). Elements
of Tail(X, φ) are not in a cycle. Rather, β is a tail element of φ if φm (β) is periodic for some integer
m ≥ 0, but φk (β) ∈ Tail(X, φ) for each integer 1 ≤ k < m. Lastly, the minimal such n is the minimal
period of α, and we define Pern (X, φ)) = {α : α has minimal period n}.
We now consider an example to see how these definitions work in practice. Consider F11 , the
finite field with 11 elements, and φ : F11 → F11 defined by φ(x) = x2 . Then, the finite dynamical
system (F11 , φ) produces the digraph in Figure 1.
Using the digraph, we easily see that Per(F11 , x2 ) = {0, 1, 3, 4, 5, 9} with Fix(F11 , x2 ) = {0, 1}.
Additionally, Tail(F11 , x2 ) = {2, 6, 7, 8, 1}. We see that φ4 (3) = 3, which means that 3 ∈ Per4 (F11 , x2 ).
Likewise, 4, 5 and 9 are also in Per4 (F11 , x2 ).
Finite Dynamical Systems
3
2. R ANDOM FINITE DYNAMICAL SYSTEMS FROM A PROBABILISTIC POINT OF VIEW
In this section, we use probability theory techniques to determine the asymptotic behavior of
random finite dynamical systems. To begin, we first note that the number of periodic points for a
function φ ∈ Dyn(Xq ) depends only on q. We have the following proposition.
Proposition 2.1. Let Xq be a finite set and let φ ∈ Dyn(Xq ). Then φ is bijective if and only if
Per(Xq , φ) = Xq .
Proof. First we prove necessity. Suppose that φ is bijective. We must show that Per(Xq , φ) = Xq .
Let α ∈ Xq . Since Xq is a finite set, all elements of Xq are preperiodic.
Thus, there exists k ∈ N
k
m
k
k
such that φ (α) is periodic. Then, for some m ∈ N, φ φ (α) = φ (α). But this implies that
φm φk (c) = φk (φm (α)) = φk (α), since φ is one-to-one. It follows that φm (α) = α, so α is a
periodic. Thus, Per(Xq , φ) = Xq .
To prove sufficiency, suppose that Per(Xq , φ) = Xq . First we show that φ is onto. Let α ∈ Xq .
Since Per(Xq , φ) = Xq , there is b ∈ N such that φb (α) = α. Note that φ(φb−1 (α) = φb (α) = α.
Thus, φ is onto.
Since Xq is a finite set and φ is onto, φ is one-to-one by the pigeonhole principle. Thus, φ is
bijective.
Thus, as the expected number of periodic points depends only on set size in the bijective case,
we focus on the general and d-to-one cases exclusively.
In the general case we first calculate the expected number of periodic points for a randomly
selected function, and then reduce this calculation to an asymptotic formula. From this calculation,
we proceed to find a asymptotic formula for variance, which leads to a later conjecture concerning
the value distribution of the number of periodic points for a set of size q under arbitrary functions.
The d-to-one case proves to be much more difficult to work in. We begin by finding an explicit
formula for the expected number of periodic points for a set of size q under a random d-to-one
function. We then provide a conjecture about this asymptotic formula. We close this section by
providing graphical evidence for our conjecture.
From this point forward, we will let Xq = {α1 , . . . , αq } denote a set with q elements. Define the
sample space Dyn(Xq ) to be the set of all functions φ : Xq → Xq , where each φ ∈ Dyn(Xq ) occurs
with equal probability. Similarly, define the sample space Dynd (Xq ) to be the set of all d-to-one
functions φ : Xq → Xq , again where each φ ∈ Dynd (Xq ) occurs with equal probability. We define a
random finite dynamical system of order q to be a function φ ∈ Dyn(Xq ), and we define a random
d-to-one finite dynamical system of order q similarly.
2.1. Reduction of Cases.
Since we calculate the expected number of periodic points for a randomly selected function in
several cases, it is easiest to provide a uniform mental framework for carrying out these calculations. We begin by defining an n-cycle in the finite dynamical system setting.
4
Beckwith and Dickens
Definition 2.2. Let Cn = (α1 , ..., αn ) be an n-tuple of distinct elements αi of Xq . We say that Cn is
an n-cycle for φ if φ(αi ) = αi+1 for each i ≤ n and φ(αn ) = α1 . We consider permutations of the
elements of Cn by shifting to be the same n-cycle.
Using this definition of an n-cycle, it becomes easier to determine the probability that a n-tuple
is an n-cycle for a random finite dynamical systems φ. Lemmas in the following sections provide
the probability that a given n-tuple is an n-cycle in various cases for a function φ.
2.2. General Case.
Before we begin more rigorous calculations, we first determine the number of functions in
Dyn(Xq ), and prove independence of events in the general case.
Let αi ∈ Xq . Since we are working in the general case, we know that α can be mapped to q other
elements. Likewise, the other q − 1 elements in Xq have q possible images in the range. Thus, for
each αi , we have q possible possibilities which means |Dyn(Xq )| = qq .
Next, fix α, α∗ , β and β∗ in Xq with α and α∗ distinct. Let φ ∈ Dyn(Xq ). We claim that
P{φ(α) = β ∧ φ(α∗ ) = β∗ } = P{φ(α) = β}P{φ(α∗ ) = β∗ }. We see that P{φ(α) = β ∧ φ(α∗ ) =
β∗ } = P{φ(α) = β}P{φ(α∗ ) = β∗ | φ(α) = β}. Since every element has q possible range elements, the event, φ(α) = β, does not effect probability that φ(α∗ ) = β∗ . Then we have P{φ(α) =
β}P{φ(α∗ ) = β∗ | φ(α) = β} = P{φ(α∗ ) = β∗ }P{φ(α∗ ) = β∗ }, as desired. Thus, we know individual mappings are independent of each other. Note that P{φ(α) = β} = q1
2.2.1. Expected number of periodic points.
In this section we calculate the expected number of periodic points for a random finite dynamical
system. We then obtain an asymptotic formula describing the average size of |Per(Xq , φ)| as q → ∞.
Lemma 2.3. Let Xq be a set. Consider α1 , . . . , αn ∈ Xq and β1 , . . . , βn ∈ Xq where αi are
distinct and βi are distinct. Define p(n, q) = P{φ(αi ) = βi for all i}. Fix n ≥ 1. Then, in the
special case where β1 = α2 , β2 = α3 , . . . , βn = α1 the probability that we have an n-cycle is
1
p(n, q) = P {φ(α1 ) = α2 ∧ φ(α2 ) = α3 ∧ · · · ∧ φ(αn ) = α1 }. In the general case, p(n, q) = n .
q
Proof. Since φ(α1 ) = β1 and φ(α2 ) = β2 are independent events for αi , βi ∈ Xq , we see that:
p(n, q) = P{φ(α1 ) = β1 ∧ φ(α2 ) = β2 ∧ · · · ∧ φ(αn ) = βn }
= P{φ(α1 ) = β1 }P{φ(α2 ) = β2 } . . . P{φ(αn ) = βn }
1 1 1
= · ···
q q q
1
= n
q .
Finite Dynamical Systems
5
With a calculation of p(n, q) in hand, we begin our computation of E[|Per(Xq , φ)|]. For each
n-tuple C of elements of X, define indicator functions fC : Dyn(Xq ) → R by
(
1 if C is an n-cycle under φ
fC (Xq , φ) :=
0 otherwise
.
Using our calculation for p(n, q) above, we note that:
E[ fC (Xq , φ)] = 0 · (1 − P{α is a n-cycle}) + 1 · P{α is a n-cycle})
= 0 + 1 · p(n, q)
1
= n
q .
Since |Pern (Xq , φ)| is the sum of the fC (Xq , φ), we can calculate the expected number of periodic
points of order n in Xq :
E[|Pern (Xq , φ)] = n
∑
E[ fc (Xq , φ)]
∑
p(n, q)
C⊆X, |C|=n
=n
C⊆X, |C|=n
q!
=n
n(q − n)!
q!
=
(q − n)!qn .
q
Finally, |Per(Xq , φ)| =
1
qn
∑ |Pern(Xq, φ)|. Thus,
n=1
q
E[|Per(Xq , φ)] =
∑ E[|Pern(Xq, φ)|]
n=1
q
=
q!
∑ (q − n)!qn
n=1
.
We now have an exact value for E[|Per(Xq , φ)|]. Since a simple relationship is not apparent between
q and E[|Per(Xq , φ)|], we find an asymptotic formula that more clearly illustrates the growth of
E[|Per(Xq , φ)|] in terms of q as q → ∞.
2.2.2. Reduction to asymptotic formulas.
√
From initial examination, it appears that E[|Per(Xq , φ)|] is asymptotic to K q for some constant
√
K. This behavior is illustrated in Figure 2 where the upper curve is q and the lower curve is
E[|Per(Xq , φ)|]. We have the following theorem:
Theorem 2.4. Let (Xq , φ) be a random finite dynamical system. Then,
r
π√
E[|Per(Xq , φ)|] ∼
q
2
6
Beckwith and Dickens
F IGURE 2. E[|Per(Xq , φ)|] versus
√
q.
as q → ∞.
Proof. We begin by rearranging the explicit formula for E[|Per(Xq , φ)|] to get an expression that
includes the Taylor series for eq truncated at the qth term:
q
E[|Per(Xq , φ)|] =
q!
∑ (q − n)!qn
n=1
= q!q−q
= q!q
−q
q
qq−n
∑
n=1 (q − n)!
q−1
qm
m=0 m! .
∑
√
We divide E[|Per(Xq , φ)|] by its anticipated asymptotic order q, and then show that this ratio
r
π
converges to
. Using Stirling’s approximation for q!, we have
2
q−1 m
q q
E[|Per(Xq , φ)|]
1 p
q
λq −q
=
2πq
e
q
√
√
∑
q
q
e
m=0 m!
=
√ λ −q q−1 qm
2πe q e ∑
m=0 m! .
for some
1
1
≤ λq ≤
,
12q + 1
12q
Finite Dynamical Systems
Since
7
1
1
≤ λq ≤
, λq → 0 as q → ∞, we have eλq → 1 as q → ∞. Hence,
12q + 1
12q
√ λ −q q−1 qm
E[|Per(Xq , φ)|]
lim
= lim 2πe q e ∑
√
q→∞
q→∞
q
m=0 m!
√
= lim 2π e−q
q→∞
=
√
2π lim e−q
q→∞
r
=
q
qm e−q qq
∑ m! − q!
m=0
!
qm √
e−q qq
−
2π
lim
∑
q→∞ q!
m=0 m!
q
π √
e−q qq
− 2π lim
q→∞ q! .
2
q
1
qm
= . It remains to be shown that
∑
q→∞
2
m=0 m!
q q p
−q
q
e q
lim
= 0. Recall that Stirling’s formula approximates q! from below; we have q! =
2πqeλq
q→∞ q!
e
q q p
1
1
2πq < q!. Therefore,
for some
≤ λq ≤
. Since λq > 0,
12q + 1
12q
e
See the appendix for proof that the first term lim e−q
e−q qq
< lim
q→∞ q!
q→∞
lim
e−q qq
q q √
2πq
e
1
= lim √
q→∞ 2πq
= 0.
e−q qq
e−q qq
e−q qq
e−q qq
, we conclude that lim
> 0 for all q > 0 and lim
< lim q q √
= 0.
q→∞ q!
q→∞ q!
q→∞
q!
2πq
e
r
r
E[|Per(Xq , φ)|]
π
π√
Finally, this means that lim
=
, so E[|Per(Xq , φ)|] ∼
q.
√
q→∞
q
2
2
Since
2.2.3. Calculation of variance.
In this section we calculate the variance of |Per(Xq , φ)| we find an asymptotic formula for the
variance as q → ∞.
Theorem 2.5. Let φ ∈ Dyn(Xq ) be a random finite dynamical system. Then,
Var(|Per(Xq , φ)|) ∼
as q → ∞.
4−π
2
8
Beckwith and Dickens
Proof. We begin by finding the second moment of |Per(Xq , φ)|:

!2 
q
E |Per(Xq , φ)|2 = E  ∑ |Pern (Xq , φ)| 
n=1
"
=E
=
q
#
q
∑ ∑ |Pern(Xq, φ)| · |Perk (Xq, φ)|
n=1 k=1
q q
∑ ∑E
n=1 k=1
|Pern (Xq , φ)| · |Perk (Xq , φ)| .
The bulk of this calculation centers around finding E |Pern (Xq , φ)| · |Perk (Xq , φ)| given n and k.
Let N and K be n− and k-tuples in Xq respectively. Define indicator variables fN,K : Dyn(Xq ) → R
for φ ∈ Dyn(Xq ) by
(
1 if N is an n-cycle and K is a k-cycle under φ
fN,K (Xq , φ) :=
0 otherwise.
Note that for each pair of n, k-cycles N and K,
 1
if N ∩ K = 0/

 q1n+k
E fN,K (Xq , φ) = qn if N = K


0 if N 6= K and N ∩ K 6= 0/
by the arguments made previously for arbitrary functions. We now compute E |Pern (Xq , φ)| · |Perk (Xq , φ)|
/ We find that:
when N ∩ K = 0.
E |Pern (Xq , φ)| · |Perk (Xq , φ)| =
∑ nk · E fN,K (Xq, φ)
C,K disjoint
n-,k-cycles,
respectively
=
∑
C,K disjoint
n-,k-cycles,
respectively
nk
1
qn+k
(q − n)!
1
q!
n+k
n(q − n)! k(q − n − k)! q
q!
=
(q − n − k)!qn+k .
= nk
Now we consider the case where N = K. Then we have Pern (Xq , φ) = Perk (Xq , φ). Using indicator variables for n-cycles as defined above, we get:
E |Pern (Xq , φ)| =
∑ n2E fN (Xq, φ)
N an n-cycle
=n
q!
.
(q − n)!qn
Finite Dynamical Systems
9
We now return to our original calculation. From our work above, now have:
bq/2c
q
q!
q!
q!
−
+
∑ ∑ (q − n − k)!qn+k ∑ (q − 2n)!q2n ∑ n (q − n)!qn
n=1 k=1
n=1
n=1
q−1 q−n
E[|Per(Xq , φ)|2 ] =
q
=
∑ (m − 1)
m=1
q
=
∑
m
m=1
bq/2c
q
q!
q!
q!
−
+
n
∑
∑
m
2n
n
(q − m)!q
n=1 (q − 2n)!q
n=1 (q − n)!q
bq/2c
q
q
q!
q!
q!
q!
−
−
+
n
∑
∑
∑
m
m
2n
n
(q − m)!q
m=1 (q − m)!q
n=1 (q − 2n)!q
n=1 (q − n)!q
bq/2c
q!
q!
−
∑ (q − m)!qm ∑ (q − 2n)!q2n + q
n=1
m=1
q
= q−
bq/2c
q!
q!
−
∑ (q − m)!qm ∑ (q − 2n)!q2n
.
m=1
n=1
q
= 2q −
q
q!
= q. Again, we find an asymptotic formula for
See the appendix for proof that ∑ n
(q − n)!qn
n=1
E |Per(Xq , φ)|2 that gives the growth of Var E[|Per(Xq , φ)|] in terms of q. As in the calculation
r
π√
of E[|Per(Xq , φ)|], the second term is asymptotic to
q. We also note that:
2
r
bq/2c
q
q!
π√
q!
q
∼
∑
≤ ∑
2n
n
n=1 (q − 2n)!q n=1 (q − n)!q
2 .
This gives us:
r
π√
2
E |Per(Xq , φ)| ∼ 2q + 2
q ∼ 2q.
2
Finally, we conclude that:
2
Var(|Per(Xq , φ)|) = E |Per(Xq , φ)|2 − E |Per(Xq , φ)|
π
∼ 2q − q
2
4−π
=
q,
2
as desired.
2.2.4. Convergence to the normal distribution.
One motive for computing the mean and variance of |Per(Xq , φ)| is the observation that the
value distribution of |Per(Xq , φ)| appears to converge in some sense to a normal distribution as q
increases. In Figure 3 we illustrate this convergence for several large values of q. We conjecture
10
Beckwith and Dickens
upon normalization that as q → ∞,
q √ 

|Per(Xq , φ)| − π2 q dist

 −−→ N (0, 1).
q
4−π √
q
2
Number of Periodic Points, |X|=5000
0.008
0.000
0
20
40
60
80
100 120
0
50
100
150
200
250
300
Periodic Points
Number of Periodic Points, |X|=10000
Number of Periodic Points, |X|=20000
0.002
0.000
0.002
Density
0.004
0.004
0.006
Periodic Points
0.000
Density
0.004
Density
0.010
0.000
Density
0.020
Number of Periodic Points, |X|=1000
0
100
200
300
400
0
Periodic Points
100
200
300
400
500
Periodic Points
F IGURE 3. Distribution of |Per(Xq , φ)| for 5000 φ ∈ Dyn(Xq ) for Xq of increasing size.
In this section we describe our ideas for proving convergence in distribution. Thus far we have
considered two routes for a proof: the first is to find asymptotic formula for the moments of
|Per(Xq , φ)| using the properties of the function Q(m, n), which gives the number of integer partitions of an integer m into exactly n distinctparts. We anticipate
that the nth moment of |Per(Xq , φ)|
q √ q
π
4−π √
is asymptotic to the nth moment of a N
q random variable. The second ap2 q,
2
proach is to obtain a probability distribution function for |Per(Xq , φ)|, and then use one of the
standard methods for proving convergence in distribution, such as showing pointwise convergence
of characteristic functions or the method of moments. We have not as of yet been successful in
using either of these approaches, but we include them here for the sake of completion.
2.2.5. Asymptotic formulas for the moments of |Per(Xq , φ)|.
Finite Dynamical Systems
11
q
q
π√
4−π √
n
n
q ,
If we can show that for each n ∈ N, E |Per(Xq , φ)| ∼ E[Y ] as q → ∞, where Y ∼ N
2 q,
2
then
|Per(Xq ,φ)|
√
q
→
Y
√
q
in distribution as q → ∞. We have
!n #
"
q
E |Per(Xq , φ)|n = E
∑ Perk (Xq, φ)
q
=
k=1
q
∑ ∑
k1 =1 k2 =1
q
=
∑
···
q
∑E
kn =1
∑
M=1 k1 +···+kn =M
|Perk 1 (Xq , φ)| · |Perk 2 (Xq , φ)| · · · |Perk n (Xq , φ)|
q
E |Perk 1 (Xq , φ)| · · · |Perk n (Xq , φ)| + ∑ E |Perk (Xq , φ)|n
k=1
+ (a bunch of other stuff)
Consider the first term, where k1 , . . . , kn are all distinct. Fix M. Define indicator variables fC1 ,...,Cn :
Dyn(Xq ) → R for φ ∈ Dyn(Xq ) by
(
1 if each Ci is a ki -cycle for φ
fC1 ,...,Cn (Xq , φ) :=
0 otherwise.
Note that for each ordered tuple of k1 -, . . . , kn -cycles C1 , . . . ,Cn , we have
1
E fC1 ,...,Cn (Xq , φ) = P(each Ci is a ki -cycle for φ) = M ,
q
since the ki sum to M. Now note that for any combination of k1 , . . . , kn with k1 + · · · + kn = M,
1
E |Perk 1 (Xq , φ)| · · · |Perk n (Xq , φ)| =
k1 · · · kn M
∑
q
all (C ,...,C )−tuples
1
n
= k1 k2 · · · kn
=
q!
(q − k1 )!
(q − k1 − · · · − kn−1 )! 1
···
k1 (q − k1 )! k2 (q − k1 − k2 )!
kn (q − k1 − · · · − kn )! qM
q!
(q − M)!qM .
Thus the first term of our first calculation is
q
∑ ∑ E |Perk 1(Xq, φ)| · · · |Perk n(Xq, φ)| =
M=1 kn +···+kn =M
q
q!
M
M=1 k1 +···+kn =M (q − M)!q
∑
∑
q
=
q!
∑ Q(M, n) (q − M)!qM
M=1
,
where Q(m, n) denotes the number of ways to partition m into exactly n distinct integer parts. There
is some documentation on Q(m, n); for example, a generating function is
n
xn+(2)
k
Fn (x) = ∑ Q(k, n)x =
.
(1 − x) · · · (1 − xn )
k≥1
12
Beckwith and Dickens
n−1
m
Additionally, (citation needed) recently showed that Q(m, n) ∼ n[(n−1)!]
2 . We have also looked at
the last term, the ‘diagonal’ term. Define indicator variables for k-cycles as before, and compute:
E |Perk (Xq , φ)|n = ∑ kn E fC (Xq , φ)
C a k-cycle
= kn−1
q!
.
(q − k)!qk
So now in view to our first calculation we have
q
q
q!
q!
n
+ ∑ kn−1
E |Per(Xq , φ)| = ∑ Q(M, n)
+ (a bunch of other stuff).
M
(q − M)!q
(q − k)!qk
M=1
k=1
The ‘other
stuff’ is a bunch of sums that
depend on whether a given ki is repeated in the expected
value E |Perk 1 (Xq , φ)| · · · |Perk n (Xq , φ)| . Based off of the calculation for the second moment, we
suspect that the asymptotic order of these terms may be less than the two terms we calculated, so
that we can ignore the ‘other stuff’ in the limit.
We looked at the several moments computationally and each seems to have asymptotic order
n/2
q , where n is the order of the moment. This conforms with our suspicion that the distribution
of |Per(Xq , φ)| is converging to the normal distribution, since the moment generating function for
N(µ, σ) random variable is
1 2 2
ψX (t) = eµt+ 2 σ t ,
and its first few moments are as provided in the table that follows.
Order
Moment
1
µ
2
2
µ + σ2
3
µ3 + 3µσ2
4
µ4 + 6µ2 σ2 + 3σ4
5
µ5 + 10µ3 σ2 + 15µσ4
!
r r
|Per(Xq , φ)|
π
4−π
∼N
,
in the limit, the moments of
In our case where we believe
√
q
2
2
√
|Per(Xq , φ)| should be asymptotic to powers of q.
2.2.6. A distribution function for |Per(Xq , φ)|.
Theorem 2.6. Let fq : N → [0, 1] be the probability distribution function for |Per(Xq , φ)|. Then fq
is given by
q!n
fq (n) = P |Per(Xq , φ)| = n =
.
(q − n)!qn+1
To obtain a distribution function for |Per(Xq , φ)|, we first need the following lemma.
Lemma 2.7. This lemma is an adaptation from [3]. Let q ∈ N. For Xq = {α1 , . . . , αq }, for n ≤ q fix
a bijective function ψ : {α1 , . . . , αn } → {α1 , . . . , αn }. Then the number of functions φ ∈ Dyn(Xq )
such that φ|{α1 ,...,αn } = ψ and Per(Xq , φ) = {α1 , . . . , αn } is given by
T (n) = nqq−n−1 .
Finite Dynamical Systems
13
Proof. (of Lemma 2.6) Proceed by induction on q. For the base case with q = 1, the only function
is φ defined by φ(x) = α1 . Hence T (1) = 1 · 11−1−1 = 1.
Now suppose that there is q ∈ N such that T (n) = nkk−n−1 for each n ≤ k for all k < q. Let n ≤ q
and fix a bijective function ψ : {α1 , . . . , αn } → {α1 , . . . , αn }. The number of functions φ : Xq → Xq
such that φ|{α1 ,...,αn } = ψ and Per(Xq , φ) = {α1 , . . . , αq } is the same as the number of functions
φ∗ : {αn+1 , . . . , αq } → Xq such that φ∗ has no cycles. Then for each αi with i ≥ n + 1, we must
have (φ∗ )m (αi ) = α j for some j ≤ n and m ∈ N. Consider the set C = (φ∗ )−1 [{α1 , . . . , αn }], the
set of all elements mapping into the cycle structure. Write N = |C|. There are q−n
ways to
N
∗
choose the elements of C. Furthermore, we must have N ≥ 1, otherwise φ would not be acyclic.
Additionally, there are nN ways to map the elements of C to {α1 , . . . , αn }. Since φ∗ must be acyclic
over the remaining set {αn+1 , . . . , αq } \ C, and N ≥ 1, we must have φ∗ [{αn+1 , . . . , αq } \ C] = C.
Regarding C as a cycle structure, then by the inductive hypothesis we have
T (N) = N(q − n)q−N−1
as the number of ways to map the elements of {αn+1 , . . . , αq } \C. Summing over N, then we get
q−n q−n N
T (q) = ∑
n N(q − n)q−N−1
N
N=1
q−n
(q − n − 1)!
nN−1 (q − n)q−N
(N
−
1)!(q
−
n
−
N)!
N=1
q−n−1 q−n−1 i
=n ∑
n (q − n)q−n−1−i
i
i=0
=n
∑
= n(q − n + n)q−n−1
= nqq−n−1 ,
as desired.
Proof. (of Theorem 2.5) Write Xq = α1 , . . . , αq . Then
fq (n) = P |Per(Xq , φ)| = n = ∑ P Per(Xq , φ) = Y
Y ⊆X
|Y |=n
q =
P Per(Xq , φ) = {α1 , . . . , αn } ,
n
since the choice of elements for {α1 , . . . , αn } is arbitrary. We next compute P Per(Xq , φ) = {α1 , . . . , αn } .
We have
φ ∈ Dyn(Xq ) : Per(Xq , φ) = {α1 , . . . , αn } P Per(Xq , φ) = {α1 , . . . , αn } =
|Dyn(Xq )|
−q =q
φ ∈ Dyn(Xq ) : φ|{α1 ,...,αn } is bijective T (n)
= q−q n!nqq−n−1
14
Beckwith and Dickens
Finally, we conclude that
q −q
q!n
fq (n) =
q n!nqq−n−1 =
.
n
(q − n)!qn+1
Interestingly, this is the same distribution function as the one we used in Proposition 4.2.
2.3. d-to-one Case.
While in the bijective case, |Per(Xq , φ)| is completely determined by q, we might expect there to
be more variation in |Per(Xq , φ)| where we restrict φ to be a d-to-one function with d ≥ 2. We find
that d-to-one functions
behave more
like
the arbitrary case, despite their additional constraints. For
q
q
a given d, there are
d-to-one functions in Dyn(Xq ), and we denote the set of
d, . . . , d m m
d-to-one functions by Dynd (Xq ).
..
.
..
.
n
..
.
..
.
m−n
q
F IGURE 4. The large unfilled circles represent the fibers of φ while the smaller
d
q
black dots represent the elements in φ[Xq ]. We choose d − 1 elements for the first
d
n fibers that map to the n cycle elements. The remaining m − n elements of φ[Xq ]
and their fibers are then chosen so that each fiber has d elements.
2.3.1. Calculation of expected number of periodic points.
Finite Dynamical Systems
15
Lemma 2.8. Let φ ∈ Dynd (Xq ). Consider α1 , . . . , αn ∈ Xq and β1 , . . . , βn ∈ Xq where αi are
distinct and βi are distinct. Define p(n, q) = P{φ(αi ) = βi for all i}. Fix n ≥ 1. Then, in the
special case where β1 = α2 , β2 = α3 , . . . , βn = α1 the probability that {α1 , . . . , αn } for an n-cycle
is p(n, q) = P {φ(α1 ) = α2 ∧ φ(α2 ) = α3 ∧ · · · ∧ φ(αn ) = α1 } = (q−d)(q−2d)...(q−(n−1)d)
q(q−1)2 (q−2)2 ...(q−n+1)2
.
Proof. Let φ ∈ Dynd (Xq ) be a d-to-one function. Consider α1 , . . . , αn ∈ Xq and β1 , . . . , βn ∈ Xq
where the αi are distinct and the βi are distinct as well. First we calculate |{φ ∈ Dynd (Xq ) :
q
φ(α1 ) = β1 ∧ · · · ∧ φ(αn ) = βn }|. Write = m.
d
We use Figure 4 to help calculate | φ ∈ Dynd (Xq ) | φ is an n-cycle for C |. First we choose
m fiber elements. Since we have already chosen n elements in the n-cycle C, we choose m − n
additional fiber elements. Excluding the n elements in C, we can choose these m − n elements
q−n
from q − n remaining elements in X. This portion of the calculation contributes
. Now
m−n
that we have chosen the fiber elements, we must fill each fiber. We will accomplish this using a multinomial coefficient to determine the possible ways to distribute q − n elements into
m groups. Note that each fiber should have d elements since φ is a d-to-one function. In Figure 4, exactly one αi from C is in the first n fibers. Since, each fiber must have d elements,
we must choose an additional d − 1 elements for the first n fibers. Since the remaining m − n
fibers do not yet have any elements in them, we must choose d elements
for each. Thus, the
q−n
total number of ways to fill m fibers is the multinomial coefficient
,
d − 1, . . . , d − 1, d, . . . , d
where we have n groups of size d − 1 and m − n groups of size d. Now compute:
p(n, q) =
=
|{φ ∈ Dynd (Xq ) : φ(α1 ) = β1 ∧ · · · ∧ φ(αn ) = βn }|
|Dynd (Xq )|
q−n
q−n d−1,...,d−1,d,...,d m−n
q
q
d, . . . ,d m m
(q − n)!(q − n)!(d!)m m!(q − m)!
((d − 1)!)n (d!)m−n (m − n)!(q − m)!q!q!
d n m![(q − n)!]2
=
(m − n)!(q!)2
q(q − d)(q − 2d) . . . (q − (n − 1)d)
= 2
q (q − 1)2 (q − 2)2 . . . (q − n + 1)2
(q − d)(q − 2d) . . . (q − (n − 1)d)
=
q(q − 1)2 (q − 2)2 . . . (q − n + 1)2 .
=
Now we begin our calculation of E[|Per(Xq , φ)|] for random φ ∈ Dynd (Xq ). Let C be a n-tuple in
X. Then the probability that C is a n-cycle under φ is p(n, q). Define functions fC : Dyn(Xq ) → R
16
Beckwith and Dickens
by
(
1 if C is an n-cycle under φ
fC (Xq , φ) :=
0 otherwise
.
We note that:
E[ fc (Xq , φ)] =
(q − d)(q − 2d) . . . (q − (n − 1)d)
q(q − 1)2 (q − 2)2 . . . (q − n + 1)2 .
The expected number of periodic points of period n for the system (Xq , φ) is then
E[|Per∗n (Xq , φ)|] = n
∑
E[ fc (Xq , φ)]
∑
p(n, q)
C⊆X, |C|=n
=n
C⊆X, |C|=n
=n
(q − d)(q − 2d) . . . (q − (n − 1)d)
q(q − 1)2 (q − 2)2 . . . (q − n + 1)2
C⊆X, |C|=n
∑
q!
(q − d)(q − 2d) . . . (q − (n − 1)d)
·
(q − n)!n q(q − 1)2 (q − 2)2 . . . (q − n + 1)2
(q − d)(q − 2d) . . . (q − (n − 1)d)
=
(q − 1)(q − 2) . . . (q − (n − 1))
= n·
n−1
(q − kd)
k=1 (q − k) .
=∏
Finally, summing over all possible cycle lengths, we find the expected number of periodic points
for (X, φ) to be
q/d n−1
E[|Per(Xq , φ)|] =
(q − kd)
n=1 k=1 (q − k) .
∑∏
2.3.2. Conjecture of asymptotic formula for expected number of periodic points.
As in the general case, the explicit formula for E[|Per(Xq , φ)|] does not illustrate a clear relationship between q and |Per(Xq , φ)|. However, providing an explicit asymptotic formula for
E[|Per(Xq , φ)|] has proven much more difficult in the d-to-one setting. Upon cursory examination
√
of Figure 5, it appears that E[|Per(Xq , φ)|] has asymptotic order q, up to a constant that is dependent on d. Indeed, in the graph on the right of Figure 6, it appears that q−1/2 E[|Per(Xq , φ)|]
approaches a constant for each d. From this experimental data, we conjecture that
r
π
√
E[|Per(Xq , φ)|] ∼
q
2(d − 1)
as q → ∞.
Finite Dynamical Systems
17
80
60
40
20
1000
2000
3000
4000
5000
F IGURE 5. E[|Per(Xq , φ)|] for d-to-one maps, with d varying between 2 and 6. It
√
appears that E[|Per(Xq , φ)|] has asymptotic order q up to some constant dependent
on d.
1.2
1.0
0.8
0.6
0.4
1000
2000
3000
4000
5000
E[|Per(Xq , φ)|]
for d-to-one maps, with d varying between 2 and 6. It
√
q
E[|Per(Xq , φ)|]
appears that
approaches a constant for each d.
√
q
F IGURE 6.
3. F INITE DYNAMICAL SYSTEMS OVER FINITE FIELDS OF PRIME ORDER
At least three natural number-theoretic settings exist when studying finite dynamical systems.
Each case is the result of certain predictable relationships between the set and map that comprise a
finite dynamical system. In the following three sections we explain relationships that define each
case.
3.1. General case for finite fields.
Not surprisingly in the general case, few restrictions exist. The only requirement of the system
is that φ is defined as a map from X to itself. As a result, describing the behavior of a randomly
18
Beckwith and Dickens
selected function is difficult as a function may map every element in X to a single element in the
range or may be bijective.
3.2. Bijective case for finite fields.
Restricting φ to be a bijective map on the set X adds a lot of structure to the finite dynamical
system. It is easy to show that that if φ is bijective from X to X, then every element of X is periodic.
Since a primary questions of interest is the expected number of periodic points for a dynamical
system, the bijective case is overly restrictive. As a result, we would like to avoid combinations
of sets and functions that make a bijective finite dynamical system. Luckily, an easy check exist
which proves whether a system is bijective.
Proposition 3.1. Let k be an integer, let c ∈ F p and define φ : F p → F p by φ(x) = xk + c. Then
Per(F p , φ) = F p if and only if gcd(k, p − 1) = 1.
Proof. To prove necessity, suppose that Per(F p , φ) = F p . Then φ is injective by Lemma 3.1. Proceed by contradiction by supposing that gcd(k, p − 1) = m ≥ 2.
p−1
Note that φ(1) = c. We know that F∗p is cyclic, so F∗p = hgi for some g ∈ F p . Take α = g m .
p−1 k
k
p−1
k
p−1
< p−1, we know α = g m 6= g p−1 = 1. However, αk = g m
Since
= g p−1 m = 1 m = 1.
m
Then we have φ(α) = c = φ(1). But α 6= 1, so this contradicts the fact that φ is one-to-one. Hence
gcd(k, p − 1) = 1.
Next we prove sufficiency. Suppose that gcd(k, p − 1) = 1. We show that Per(F p , φ) = F p by
proving φ is bijective. We begin by showing that φ is one-to-one. Since gcd(k, p − 1) = 1, there
exist x, y ∈ Z such that xk + y(p − 1) = 1. Suppose that φ(α) = φ(β). Then, αk = βk . Since
xk + y(p − 1) = 1, we have xd = 1 − y(p − 1). Hence:
αk = β k
αxk = βxk
α1−y(p−1) = β1−y(p−1) .
Since y(p − 1) ≡ 0 (mod p − 1), we have α = β. Thus, φ is one-to-one.
Now we show that φ is onto. Let α ∈ F p . Write β = (α − c)x . We claim that φ(β) = α. Since
α, c ∈ Fp and x ∈ Z, it is easy to see that β = (α − c)x ∈ F p . We have that φ(β) = φ ((α − c)x ) =
[(α − c)x ]d + c = (α − c)1−y(p−1) + c = (α − c)1 + c = α. Thus, φ is onto. Since φ is a bijection, it
follows from Lemma 3.1 that Per(F p , φ) = F p .
3.3. d-to-one case for finite fields.
The combination of some sets and functions may produce d-to-one dynamical systems. These
systems have more structure than in the general case, while not making calculations trivial. Once
Finite Dynamical Systems
19
again, there is an easy way to determine whether a given finite dynamical system has d-to-one
structure. First we have the following theorem and corollary from [2].
Theorem 3.2. Let f (x) = xn + an−1 xn−1 + · · · + a0 be a polynomial. Then the congruence f (x) ≡ 0
(mod p) has precisely n distinct solutions if and only if f (x) divides x p − p (mod p).
This theorem is a fairly standard fact in elementary number theory. See for a proof. Corollary
3.3 follows immediately from Theorem 3.2.
Corollary 3.3. If r|p − 1, then xr ≡ 1 (mod p) has precisely r distinct solutions modulo p.
Proposition 3.4. Let k be an integer, let c ∈ F p and define φ : F p → F p by φ(x) = xk + c. Let
d = gcd(k, p − 1). Then φ ∈ Dyn(F p ) is a d-to-one map over F p \ {c}.
Proof. Let α ∈ φ[F p ]. Then for some x ∈ F p , φ(x) = α. Since gcd(k, p − 1) = d, k = dn for some
n ∈ N. Define y = xn . Then,
xk + c = α
xk − (α − c) = 0
yd − (α − c) = 0
(α − c)−1 yd − 1 = 0.
We will to show (α − c)−1 = zd for some z ∈ F p . Let φ(β) = α with β ∈ F p . Then φ(β) =
which implies that:
βk + c = α,
βk − c = α
βk = α − c
β−k = (α − c)−1
β−nd = (α − c)−1
(β−n )d = (α − c)−1
.
Since β ∈ F p , β−1 ∈ F p , which implies that β−n ∈ F p . Define ỹ = β−n y, so that (α − c)−1 yd =
(β−n )d yd = ỹ d . It remains to be shown that ỹd | ỹ p − ỹ. By Corollary 3.3, ỹd − 1 | ỹ p − ỹ, so by
Theorem 3.2, ỹ − 1 has d distinct roots in F p . But, ỹ − 1 = xk − (α − c) = 0, so xk − (α − c) has d
distinct roots as well, which implies that α has d preimages. Thus, φ is d-to-one over F p \ {c}. The predictions made for functions over arbitrary sets of varying size are supported by studying
the behavior of polynomial functions over finite fields. Recall that φ : F p → F p defined by φ(x) =
xk + c is d-to-one if and only if gcd(k, p − 1) = 1. When k is prime, we must have p ≡ 1 (mod k)
20
Beckwith and Dickens
in order for φ to be d-to-one. In this section we are interested in the dynamics of a fixed d-toone function φ(x) = xk + c as p → ∞. For example, normalizing |Per(φ, F p )| by the size of the
underlying field, we predict that


r


|Per(φ,
F
)|
1
π
p

lim  ∑
=
√

x→∞ 
p
| {p ≤ x : p ≡ 1 (mod k)} |
2(d − 1) .
p≤x
p≡1 (mod k)
When averaging |Per(X, φ)|, we may fix either the underlying set or the function in the dynamical
system. In our previous calculations using probability theory alone, we fixed a set size and then
found the size of the periodic set under various functions. In the present case, we fix a single
polynomial and allow the size of the underlying field to increase and observe the distribution of
|Per(φ, F p )|
normalized periodic sets. Interestingly,
appears to be normally distributed in the limit.
√
p
See Figure 7 for an illustration.
0.4
0.0
0.2
Density
0.6
0.8
Histogram of Normalized Periodic sets for (F_p,x^3+52), p<10000) p cong 1(3)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
SD
F IGURE 7. |Per(φ)| for φ(x) = x3 + 52 over F p , with p ≡ 1 (mod 3) and p < 20000.
More generally, consider a function φ : F p → F p given by φ(x) = xk + c, where k is not necessarily prime. Let a ∈ Z with gcd(a, k) = 1, and suppose that gcd(k, p − 1) = d. We conjecture
that
r
|Per(φ, F p )|
π
x
∼
√
∑
p
2(d − 1) φ(k) log x .
p≤x
p≡a (mod k)
Finite Dynamical Systems
21
100
0
50
Frequency
150
Histogram of Normalized Periodic sets for (F_p,x^4+52), p<20000) p cong 3(4)
0
1
2
3
4
pp52
F IGURE 8. |Per(φ)| for φ(x) = x4 + 52 over F p , with p ≡ 3 (mod 4) and p < 20000.
4. A PPENDIX
Proposition 4.1. Let q ∈ N. Then
lim e−q
q→∞
q
qn 1
∑ = 2.
n=0 n!
Proof. This proof closely follows an exercise in [1]. Let (Xi ) be a sequence of independent
and identically distributed random variables with each Xi ∼ Poisson(λ). For each Xi , we have
n
k
−λ λ
. Define Sn = ∑ Xi . By the additivity of Poisson random variables, Sn ∼
P {Xi = k} = e
k!
i=1
Poisson(nλ). By the central limit theorem, it follows that
Z 0
2
Sn − nλ
1
1
√
√ e−x /2 dx = .
P
<0 →
2
−∞ 2π
nλ
22
Beckwith and Dickens
We now rearrange to get the desired summation. We have
Sn − nλ
√
P
< 0 = P {Sn < nλ}
nλ
= ∑ P {Sn = k}
k<nλ
=
∑
e−nλ
k<nλ
−n
Taking λ = 1 yields e
n
nk
(nλ)k
k!
1
∑ k! → 2 .
k=1
Proposition 4.2. Let q ∈ N. Then
q
q!n
∑ (q − n)!qn = q.
n=1
Proof. Consider the sample space S = (a0 , . . . , aq ) : 1 ≤ a j ≤ q , where each ordered tuple (a0 , . . . , aq )
is chosen with equal probability. Note that |S| = qq+1 . Consider the random variable X = f ((a0 , . . . , aq )),
where f ((a0 , . . . , aq )) is the smallest 1 ≤ n ≤ q such that an = a j for some 0 ≤ j < n. Then
q(q − 1) · · · (q − (n − 1))nqq−n
qq+1
(q − 1)!n
=
(q − n)!qn .
Since X can take on values only from 1 to q,
P {X = n} =
q
1=
q
∑ P {X = n} =
n=1
Multiplying by q, we get that
q
(q − 1)!n
∑ (q − n)!qn
n=1
.
q!n
∑ (q − n)!qn = q.
n=1
R EFERENCES
[1] Mark Kac. Statistical independence in probability, analysis and number theory. The Carus Mathematical Monographs, No. 12. Published by the Mathematical Association of America. Distributed by John Wiley and Sons, Inc.,
New York, 1959.
[2] Abhinav Kumar. Lecture notes for 18.781. unpublished lecture notes.
[3] Dennis Walsh. Notes on acyclic function digraphs. unpublished lecture notes.
K ENYON C OLLEGE , S T. O LAF C OLLEGE
E-mail address: [email protected], [email protected]