@
Appl. Gen. Topol. 21, no. 2 (2020), 215-233

doi:10.4995/agt.2020.12929

c© AGT, UPV, 2020

The class of simple dynamical systems

K. Ali Akbar

Department of Mathematics, Central University of Kerala, Kasaragod - 671320, Kerala, In-

dia. (aliakbar.pkd@gmail.com, aliakbar@cukerala.ac.in)

Communicated by F. Balibrea

Abstract

In this paper, we study the class of simple dynamical systems on R
induced by continuous maps having finitely many non-ordinary points.
We characterize this class using labeled digraphs and dynamically in-
dependent sets. In fact, we classify dynamical systems up to their
number of non-ordinary points. In particular, we discuss about the
class of continuous maps having unique non-ordinary point, and the
class of continuous maps having exactly two non-ordinary points.

2010 MSC: 54H20; 26A21; 26A48.

Keywords: special points; non-ordinary points; critical points; order con-
jugacy; order isomorphism; labeled digraph; dynamically inde-
pendent set.

1. Introduction

A dynamical system is a pair (X, f), where X is a metric space and f
is a continuous self map on X. Two dynamical systems (X, f) and (Y, g)
are said to be topologically conjugate (or simply conjugate) if there exists a
homeomorphism h : X → Y (called topological conjugacy) such that h ◦ f =
g ◦ h. We simply say that f is conjugate to g, and we write it as f ∼ g. In the
case when h happens to be an increasing homeomorphism (for example, when
X = R or an interval) we say that f and g are increasingly conjugate or order
conjugate. When we are working with a single system, any self conjugacy can
utmost shuffle points with same dynamical behavior. Therefore a point which
is unique up to its behavior must be fixed by every self conjugacy. On the

Received 02 January 2020 – Accepted 17 May 2020

http://dx.doi.org/10.4995/agt.2020.12929


K. Ali Akbar

other hand if a point is fixed by all self conjugacies then it must have a special
property (some times it may not be known explicitly). These things motivated
to call the set of all points fixed by all self conjugacies as set of special points.
For x, y ∈ R, we write x ∼ y if x and y have the same dynamical properties in
the dynamical system (R, f). Said precisely, x ∼ y if there exists an increasing
homeomorphism h : R → R such that h ◦ f = f ◦ h and h(x) = y. It is easy to
see that ∼ is an equivalence relation. Since the equivalence relation is coming
from self conjugacy it is important in the field of topological dynamics. Let [x]
denote the equivalence class of x ∈ R. Let I, J be two subintervals of R. We
say that I < J if x < y for all x ∈ I and y ∈ J.

The properties of dynamical systems which are preserved by topological
conjugacies are called dynamical properties. The points which are unique up to
some dynamical property are called dynamically special points. Said differently,
a special point has a dynamical property which no other point has. We say
that a point x is ordinary if, its “like” points near it. That is, an element x ∈ R
is ordinary in (R, f) if its equivalence class [x] is a neighbourhood of it. i.e, the
equivalence class of x contains an open interval around x. A point which is not
ordinary is called non-ordinary. The idea of special points and non-ordinary
points are relatively new to the literature (see [1], [2], [8]). Recently, we studied
the class of simple dynamical systems induced by homeomorphisms. The reader
may refer [2] to get an idea of simple systems induced by homeomorphisms
having finitely many non-ordinary points. Throughout this paper, we will be
working with continuous self maps of the real line. Since R has order structure,
we would like to consider the conjugacies preserving the order. Hence the
conjugacies which we consider in this paper are order preserving conjugacies
(increasing conjugacies). For any continuous map f : R → R, we denote Gf↑
for the set of all order conjugacies of f.

In a dynamical system (X, f), let N(f) denote the set of all non-ordinary
points of f. Observe that a point to be special if [x] = {x}. Let S(f) denote
the set of all special points of f. A point x in a topological space X is said
to be rigid if it is fixed by every self homeomorphism of X. For example, the
point 1 is rigid in (0, 1]. According to the above definition all rigid points are
special, even though there is no role for the map f. By definition, the points of
[x] are dynamically the same. We consider the systems for which there are only
finitely many equivalence classes. This means there are only finitely many kinds
of orbits up to conjugacy. If f : R → R is continuous and Per(f) is properly
contained in {1, 2, 22, ...}, then f is not Li-Yorke chaotic (see [7]). Also note
that, if f : R → R is Devaney chaotic then 6 ∈ Per(f) (see [5]). Therefore, if
f : R → R is a continuous map having finitely many non-ordinary points then
neither it is Li-Yorke chaotic nor it is Devaney chaotic because of Sharkovskii’s
theorem. For these reasons we call such systems as simple systems. These are
the system in which the phase portrait can be drawn. Phase portraits (see [4])
are frequently used to graphically represent the dynamics of a system. A phase
portrait consists of a diagram representing possible beginning positions in the
system and arrows that indicate the change in these positions under iteration

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 216



The class of simple dynamical systems

of the function. The drawable systems are interesting to physicist and for this
reason the study of the class of simple dynamical systems can be useful.

Our main results (see Section 3) prove that

(i) There are exactly 26 continuous maps on R with exactly one non-
ordinary point and 90 continuous maps on R with exactly two non-
ordinary points up to order conjugacy.

(ii) Let C be the class of all continuous self maps of R, having finitely many
non-ordinary points. Then
(a) for every member of C, there exists a maximal dynamically inde-

pendent set that is a finite union of intervals.
(b) two members of C are order conjugate if and only if they have

order isomorphic maximal dynamically independent set as in (a),
and with isomorphic labeled digraphs.

(c) every order isomorphism between such maximal dynamically in-
dependent set extends uniquely to an order conjugacy.

2. Basic Results

In this Section, we will be discussing some basic results that will be used to
prove our main theorems. Most of the results are available in [2].

Let (X, f) be a dynamical system. By the full orbit of a point x ∈ X we
mean the set O(x) = {y ∈ X : fn(x) = fm(y) for some m, n ∈ N}.

For any subset A ⊂ R, let

O(A) =
!

x∈A
O(x) =

!

x∈A
{y ∈ R : fn(y) = fm(x) for some m, n ∈ N}.

A point x in a dynamical system (X, f) is said to be a critical point if f fails
to be one-to-one in every neighbourhood of x. The set of all critical points of
f is denoted by C(f).

See [2] and [8], for the following characterization theorem for the set N(f)
(and hence for S(f)).

Theorem 2.1. For continuous self maps of the real line R, the set of all non-
ordinary points is contained in the closure of the union of full orbits of critical
points, periodic points and the limits at infinity (if they exist and are finite).

Remark 2.2. In Theorem 2.1, the inclusion can be strict.

Proof. Consider the map f(x) = x + sinx for all x ∈ R. All integral multiples
of π are fixed points for this map but the increasing bijection x &→ x + 2π
commutes with f and fixes none of them. □
Remark 2.3 ([8]). For polynomials of even degree the equality is true in Theo-
rem 2.1.

For a dynamical system (X, f), let D = O(C(f) ∪ P(f) ∪ {f(∞), f(−∞)})
where f(∞) and f(−∞) are the limits of f at ∞ and −∞ respectively, provided
they are finite. For any subset A of X, we denote Ā for the closure of A in X.

Now we consider the following theorem:

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K. Ali Akbar

Theorem 2.4 ([8]). For polynomial maps f of R, S(f) has to be either empty
or a singleton or the whole D.

From the definition, it is clear that the set of special points S(f) is always
closed. The following theorem is about the converse and it is proved in [8].

Theorem 2.5. Given any closed subset F of R, there exists a continuous map
f : R → R such that S(f) = F.

Remark 2.6. For every closed subset F of R there exist continuous maps f :
R → R and g : R → R such that S(f) = Fix(g) = F . Conversely, for every
closed subset F of R there exist continuous maps f, g : R → R such that
S(f) = Fix(g) = F .

Proof. For every closed subset F of R there exists a strictly increasing continu-
ous bijection f : R → R such that Fix(f) = F . This is because, we can define
f(x) = x+ 1

2
d(x, F). Hence the remark follows from Theorem 2.5 since Fix(f)

is closed. □
The following total order on N is called the Sharkovskii’s ordering:
3 ≻ 5 ≻ 7 ≻ 9 ≻ ... ≻ 2 × 3 ≻ 2 × 5 ≻ 2 × 7 ≻ ...
≻ 2n × 3 ≻ 2n × 5 ≻ 2n × 7 ≻ ...
...2n ≻ .... ≻ 22 ≻ 2 ≻ 1
We write m ≻ n if m precedes n (not necessarily immediately) in this order.

An n-cycle means a cycle of length n.

Theorem 2.7 (Sharkovskii’s Theorem [6]). Let m ≻ n in the Sharkovskii’s
ordering. For every continuous self map of R, if there is an m-cycle, then
there is an n-cycle.

For any f : R → R, let Gf denote the set of all topological conjugacies of f
and let Gf↑ denote the set of all order conjugacies of f.

Proposition 2.8 ([2]). If x is an ordinary point of f and if h is a self topo-
logical conjugacy of f, then h(x) is ordinary.

Proposition 2.9 ([2]). If x is a non-ordinary point of f and if h is a self
topological conjugacy of f, then h(x) is non-ordinary.

Now we ask: For a continuous map f : R → R, how the equivalence classes
looks like?

The following known lemma answer this question.

Lemma 2.10 ([2]). Let f : R → R be continuous. Suppose a < b and (a, b) ∩
N(f) = ∅. Then x ∼ y for all x, y ∈ (a, b).

Theorem 2.11 ([2]). Let f : R → R be continuous. If |N(f)| = n then
|{[x] : x ∈ R}| = 2n + 1.

Remark 2.12 ([2]). Note that, being a point in a particular equivalence class [x]
is a dynamical property. There are continuous maps f : R → R having finitely
many equivalence classes, but infinitely many non-ordinary points.

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The class of simple dynamical systems

Remark 2.13.

(1) If f : R → R has a unique fixed point then it is non-ordinary and vice
versa.

(2) If f : R → R has finitely many fixed points (critical points) then all
fixed (critical) points are special and hence non-ordinary.

(3) If there are only finitely many periodic cycles then all periodic points
are special.

(4) Every special point is non-ordinary. But every non-ordinary point may
not be special.

Proof. (1) Since the topological conjugacies carry fixed points to fixed points,
the unique fixed point must be fixed by every self conjugacy and hence special.

Suppose x0 ∈ R is the unique non-ordinary point of f. Then h(x0) = x0
for all h ∈ Gf↑. Now, for any h ∈ Gf↑ we have h(f(x0)) = f(h(x0)) = f(x0).
That is, the point f(x0) is special. Since x0 is the only special point, we have
f(x0) = x0.

(2) It follows from the fact that under a topological conjugacy fixed points
will be mapped to fixed points and critical points will be mapped to critical
points and the fact that it takes the finite set of fixed points (critical points)
to itself bijectively, preserving the order.

Proof of (3) is easy.
(4) It is immediate from the definition that every special point is non-

ordinary. For the converse, consider the map x &→ x + sin x on R which has
countably many fixed points. Note that all the fixed points are non-ordinary
and they form two distinct equivalence classes, hence they are not special. □

The following proposition says that the class of maps with finitely many
non-ordinary points the idea of special points and the idea of non-ordinary
point, coincide.

Proposition 2.14. If f : R → R has only finitely many non-ordinary points
then every non-ordinary point is special.

Proof. Since N(f) is finite, it follows from proposition 2.9 that h(N(f)) = N(f)
for all h ∈ Gf↑. Then we must have h(x) = x for all x ∈ N(f), because of the
order preserving nature of h. Hence all points of N(f) are special. □

Proposition 2.15 ([2]). For maps f : R → R with finitely many non-ordinary
points, f(x) is non-ordinary whenever x ∈ R is non-ordinary.

Definition 2.16. For any subset A of R, we write ∂A = A ∩ (X − A) and call
it the boundary of A boundary of a set.

Recall that the properties which are preserved under topological conjugacies
are called dynamical properties. Hence, if two points x, y in the dynamical
system (X, f), differ by a dynamical property, then no conjugacy can map one
to the other, from which it follows that,

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K. Ali Akbar

Proposition 2.17. For any dynamical property P, the points of ∂SP are non-
ordinary where SP denotes the set of all points in (X, f) having the dynamical
property P.

Corollary 2.18. Let f : R → R be constant in a neighbourhood of a point x0.
Then the end points of the maximal interval around x0 on which f is constant,
are non-ordinary.

Recall that, if f : R → R has a unique non-ordinary point then it is a fixed
point. Next we consider the following proposition. See [2] for a proof.

Proposition 2.19 ([2]). Let f : R → R be a continuous map. Then,
(i) If x ∈ R is both critical and ordinary then f is locally constant at x.
(ii) If x is ordinary and f is not locally constant at x then f(x) is ordinary.

Remark 2.20 ([2]). Let f : R → R be continuous. Then sup f(R), inf f(R),
limx→∞ f(x) and limx→−∞ f(x) are special (in particular, non-ordinary) pro-
vided they are finite. (Note that, for maps with finitely many non-ordinary
points both limx→∞ f(x) and limx→−∞ f(x) always exists in R ∪ {−∞, ∞}).

Proposition 2.21 ([2]). The maps x + 1 and x − 1 on R are topologically
conjugate; but not order conjugate.

Proof. The maps x + 1 and x − 1 are conjugate to each other through −x + 1
2
.

If possible, let h be an order conjugacy from f(x) = x + 1 to g(x) = x − 1.
Then h(x + 1) = h(f(x)) = g(h(x)) = h(x) − 1. i.e, h(x + 1) − h(x) = −1 < 0.
Which is a contradiction to the assumption that h is increasing. □

Remark 2.22. Note that for the map x + 1 on R, all points are ordinary. For,
if a, b ∈ R, then the map x + b − a is the order conjugacy of x + 1 which maps
a to b.

Recall that for any subset A of a metric space X,

(∂A)c = int(A) ∪ int(Ac).
That is, the complement of ∂A is the union of the interior of A and the interior
of ’A complement’.

The following proposition gives a characterization for the non-ordinary points
of increasing homeomorphisms.

Proposition 2.23 ([2]). Let f : R → R be an increasing bijection and let
x ∈ R. Then x is non-ordinary if and only if x is in the boundary of Fix(f).

Next we consider:

Proposition 2.24 ([2]). Let f : R → R be a homeomorphism without fixed
points.

(i) If f(0) > 0 then f is order conjugate with x + 1.
(ii) If f(0) < 0 then f is order conjugate with x − 1.

We denote graph(f) for the range of the map f.

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The class of simple dynamical systems

Corollary 2.25 ([2]). Let f, g : (a, b) → (a, b) be homeomorphisms without
fixed points. Then f is order conjugate to g if and only if both graph(f) and
graph(g) are on the same side of the diagonal.

In particular,
(i) If f(x) > x for all x ∈ (a, b) then f is order conjugate to x + 1.
(ii) If f(x) < x for all x ∈ (a, b) then f is order conjugate to x − 1.

Remark 2.26. In fact, in the previous corollary, the interval (a, b) can be re-
placed by any open ray in R. For an increasing bijection f : R → R with
finitely many non-ordinary points, all non-ordinary points are fixed points.

3. Class of continuous maps

We now consider the systems for which there are only finitely many equiva-
lence classes. Recall that, if SP denote the set of all points having the dynamical
property P then the points of ∂SP (the boundary of SP ) are non-ordinary. As
in Remark 2.12, being a point in a particular equivalence class is a dynam-
ical property of the point. Hence by the very nature of the order conjuga-
cies, it follows that when there are finitely many non-ordinary points (there-
fore special points) there are only finitely many equivalence classes. Hence if
f : R → R is an increasing bijections with finitely many non-ordinary points
x1 < x2 < ... < xn for some n ∈ N, then these points gives rise to an ordered
partition {(−∞, x1), (x1, x2), ..., (xn, ∞)} of R \ {x1, x2, ..., xn}. This partition
gives rise to a word w(f) over {A, B, O} of length n + 1 by associating A to
f(t) > t ∀ t, B to f(t) < t ∀ t and O to f(t) = t ∀ t. We now study, the class of
simple systems induced by continuous maps having finitely many non-ordinary
points.

We first state the following results without proof. For a proof refer [2]. These
results are easy to prove and it will be used to prove our main theorems.

Proposition 3.1. Let f, g be two increasing bijections on R with finitely many
(same number of) non-ordinary points. Then f and g are order conjugate if
and only if w(f) = w(g).

Proposition 3.2. There is a one to one correspondence between the set of all
increasing continuous bijections (up to order conjugacy) on R with exactly n
non-ordinary points and the set of all words of length n + 1 on three symbols
A,B,O such that OO is forbidden.

Theorem 3.3. The number of all increasing continuous bijections (up to order
conjugacy) on R with exactly n non-ordinary points is equal to an where an =
C1(1 +

√
3)n + C2(1 −

√
3)n, C1 =

(3
√
3+5)

2
√
3

and C2 =
(3

√
3−5)

2
√
3

.

Proposition 3.4. Two decreasing bijections f and g are order conjugate (re-
spectively topologically conjugate) if and only if f2|[a,∞) and g2|[b,∞) are order
conjugate (respectively topologically conjugate) where a and b are the fixed points
of f and g respectively.

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K. Ali Akbar

Proposition 3.5. Two decreasing bijections f and g are order conjugate (re-
spectively topologically conjugate) if and only if f2|(−∞,a] and g2|(−∞,b] are or-
der conjugate (respectively topologically conjugate) where a and b are the fixed
points of f and g respectively.

Proposition 3.6. If f is a decreasing bijection from R to R with fixed point a.
Then f has 2n + 1 non-ordinary points if and only if (f ◦ f)|(a,∞) : (a, ∞) →
(a, ∞) has n non-ordinary points.

Theorem 3.7. If sn denotes the number of decreasing homeomorphisms up to
order conjugacy, then

sn =

"
0 if n is even
a n−1

2
if n is odd

for all n.

In subsections 3.1 and 3.2, we discuss the class of continuous maps with
unique (respectively two) non-ordinary points. In the Subsection 3.3, we con-
sider the class of continuous maps with finitely many non-ordinary points.

3.1. Class of continuous maps having unique non-ordinary point.

We consider the following propositions that will help us to prove our main
theorem.

Proposition 3.8. Let f, g : R → R be such that
(1) f(0) = g(0) = 0.
(2) f|(0,∞), g|(0,∞) : (0, ∞) → (0, ∞) are increasing bijections.
(3) f|(−∞,0), g|(−∞,0) : (−∞, 0) → (0, ∞) are decreasing bijections.

Then f is order conjugate to g if and only if f|(0,∞) is order conjugate to
g|(0,∞).

Proof. Suppose h : (0, ∞) → (0, ∞) is an order conjugacy from f|(0,∞) to
g|(0,∞). For x < 0, define h(x) = (g|(−∞,0))−1hf(x), and h(0) = 0. □

Proposition 3.9. Let f, g : R → R be such that
(1) f(0) = g(0) = 0.
(2) f|(−∞,0), g|(−∞,0) : (−∞, 0) → (−∞, 0) are increasing bijections.
(3) f|(0,∞), g|(0,∞) : (0, ∞) → (−∞, 0) are decreasing bijections.

Then f is order conjugate to g if and only if f|(−∞,0) is order conjugate to
g|(−∞,0).

Proof. Suppose h : (−∞, 0) → (−∞, 0) is an order conjugacy from f|(−∞,0) to
g|(−∞,0). For x > 0, define h(x) = (g|(0,∞))−1hf(x), and h(0) = 0. □

We denote the symbol
#

for disjoint union.

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The class of simple dynamical systems

Proposition 3.10.

(1) Let f : (−∞, 0] → (−∞, 0] be an increasing bijection (it follows that
f(0) = 0).
(a) If f(x) > x for all x ∈ (−∞, 0) then f is order conjugate to x

2
.

(b) If f(x) < x for all x ∈ (−∞, 0) then f is order conjugate to 2x.
(2) Let f : [0, ∞) → [0, ∞) be an increasing bijection (it follows that f(0) =

0).
(a) If f(x) > x for all x ∈ (0, ∞) then f is order conjugate to 2x.
(b) If f(x) < x for all x ∈ (0, ∞) then f is order conjugate to x

2
.

(3) Let f : [1, ∞) → [1, ∞) be an increasing bijection (it follows that f(1) =
1).
(a) If f(x) > x for all x ∈ (1, ∞) then f is order conjugate to 2x − 1.
(b) If f(x) < x for all x ∈ (1, ∞) then f is order conjugate to x+1

2
.

Proof. (1). Proof of (a):
Let f : (−∞, 0] → (−∞, 0] be an increasing bijection satisfying f(x) >

x for all x < 0. It follows that f(0) = 0. Note that for any such map#
n∈Z[f

n(x), fn+1(x)) = (−∞, 0) for all points x ∈ (−∞, 0). Then f is topo-
logically conjugate to the map x/2. We construct a topological conjugacy
h : (−∞, 0] → (−∞, 0]. For this, take any point other than 0, say −1, in the
domain. We take an arbitrary increasing homeomorphism h from [−1, f(−1))
to [−1, −1/2). Then as noted above,

#
n∈Z[f

n(−1), fn+1(−1)) = (−∞, 0).
That is, for every x ∈ (−∞, 0), there exists a unique n0 ∈ Z such that
fn0(x) ∈ [−1, f(−1)). We define h(x) = 2n0h(fn0(x)). This is well defined. It
is an increasing homeomorphism from (−∞, 0) to (−∞, 0). This h commutes
with f. This h is a conjugacy from f to the map x/2.

Similarly, we can prove (1).b, (2) and (3). □

Theorem 3.11 ([1]). There are exactly 26 maps on R with a unique non-
ordinary point, up to order conjugacy.

Proof. By Corollary 3.15, Propositions 3.8, 3.9, 3.10 (1), and 3.10 (2), the
theorem follows. □

Definition 3.12. Let f : R → R be a continuous map. Let I ⊂ R be an
interval such that fk(I) = I, and fm(I) ∕= I for all 1 ≤ m < k. Then we say
that I → f(I) → f2(I) → ... → fk−1(I) → I is a k-cycle through I. Cycles
through I, J are said to be distinct if fm(I) ∕= fn(J) for all m, n ∈ N.

We can represent each member of this class as a graph as follows:
Let f : R → R be a continuous map with a unique non-ordinary point.

Now, let I1, I2 be non singleton equivalence classes such that I1 < I2. Define
a labeled digraph (G, Vf ) with vertex set Vf = {I1, I2}, an edge from Ij to
Im if f(Ij) = Im, and a symbol I (respectively D) on this edge whenever f is
increasing (respectively decreasing) on Ij. Label the symbols A, B, O on the
least vertex of each k-cycle depends on the graph of fk, k = 1, 2 which is above
the diagonal or below the diagonal or on the diagonal respectively.

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K. Ali Akbar

Note: If f is a decreasing homeomorphism then we can consider the labeled
digraph of f2 instead of the labeled digraph f (see Proposition 3.4).

3.2. Class of continuous maps having exactly two non-ordinary points.

Next we consider the following proposition.

Proposition 3.13. Let f : R → R, g : R → R be two continuous maps having
finitely many non-ordinary points such that N(f) = N(g) and let N(f)

c
=#

Ik. For m, n ∈ N, there exist increasing bijections hn : Īn → Īn and hm :
Īm → Īm such that g ◦ hn(x) = hm ◦ f(x) for all x ∈ Īn whenever f(Īn) = Īm
and g(Īn) = Īm then f ∼ g.

Proof. Define h : R → R by
h(x) = hn(x)

for all x ∈ Īn and n ∈ N. Then h is an increasing bijection such that h◦f = g◦h.
Hence the proposition. □

Note that if a continuous bijection has finitely many non-ordinary points
then we can assume that these points are arbitrary. Because, let a1 < a2 <
... < an be the non-ordinary points of a continuous map f : R → R. Let
b1 < b2 < ... < bn be arbitrary points in R. Let h : R → R be an increasing
bijection such that h([ai, ai+1]) = [bi, bi+1] for i = 1, 2, ..., n − 1. Then define
g = h ◦ f ◦ h−1. We can easily verify b1, b2, ..., bn are the only non-ordinary
points of g since h is an order conjugacy from f to g.

The following theorem help us to classify the class of continuous maps having
finitely many non-ordinary points.

Theorem 3.14. Let f, g : R → R be two continuous maps having finitely many
non-ordinary points such that N(f) = N(g). Let N(f)c =

#
n In.

(1) If f and g have the same type of monotonicity (i.e., either both f and g
are increasing or both are decreasing) in the closure of each equivalence
class and contains exactly n distinct cycles of length ki through Ijki for

i = 1, 2, ..., n and for some jki. Then f ∼ g whenever graph (gki|Ījki
)

and graph (gki|Ījki
) are in the same side of the diagonal.

(2) If f and g have the same type of monotonicity in the closure of each
equivalence class and does not contain any cycle then f ∼ g.

Proof. For simplicity we consider the case whenever f and g have exactly one
cycle through its equivalence classes.

Let f and g have the same type of monotonicity in the closure of each
equivalence class and contain exactly one cycle through its equivalence classes
(say of length k and through their equivalence class Ij for some j).

Claim: f ∼ g whenever graph (fk|Īj ) and graph (g
k|Īj ) are same side of the

diagonal.

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The class of simple dynamical systems

Let Ij = J1 → J2 = f(J1) → ... → Jk = fk−1(J1) → J1 be the k-cycle.
Given that graph (fk|Īj) and graph (gk|Īj) are same side of the diagonal. Then
there exists an increasing bijection h : J̄1 → J̄1 such that
(3.1) fk ◦ h = h ◦ gk

Choose h1 = h. Find hi : J̄i → J̄i for i = 2, 3, ..., k and h′1 : J̄1 → J̄1 such
that f|J̄i ◦ hi = hi+1 ◦ g|J̄i and f ◦ hk = h

′
1 ◦ g. Recursively, we can prove that

h′1 = f
k ◦ h1 ◦ g−k. This implies h′1 = h by equation (3.1). Define sm(x) = x

for all x ∈ Im whenever f, g are constant on Im. In all the other equivalence
classes choose h arbitrary and define h′ (or vice versa) such that f ◦ h = h′ ◦ g.
This gives a well defined h : R → R such that h(x) = sn(x) where sn : In → In
is a continuous increasing bijection obtained as above such that f ◦sn = sm ◦g.
This implies f ◦ h = h ◦ g. Similarly we can prove the general case of (1) and
the case (2). □
Corollary 3.15. Let f, g be two increasing bijections having finitely many fixed
points such that Fix(f) = Fix(g) and let Fix(f)

c
=

#
In. If f|Īn ∼ g|Īn for

every n then f ∼ g.
Remark 3.16. Note that the complement of Fix(f) is a countable union of
open intervals (including rays) whose end points are fixed points. Since f is
increasing and the end points are fixed, no point in a component interval can
be mapped to a point in any other component interval by f.

Definition 3.17. Let A, B ⊂ R, A ∕= B, and f, g : A → B be continuous
maps. We say that f is order conjugate to g if there exist increasing bijections
hf : A → A and hg : B → B such that f ◦ hf = hg ◦ g.

Now we consider:

Proposition 3.18.

(1) Let f : (−∞, 0] → [0, 1) be a decreasing bijection (it follows that f(0) =
0). Then f is order conjugate to −x−x+1.

(2) Let f : [1, ∞) → [0, 1) be an increasing bijection (it follows that f(1) =
0). Then f is order conjugate to x−1

x
.

(3) Let f : [1, ∞) → (0, 1] be a decreasing bijection (it follows that f(1) =
1). Then f is order conjugate to 1

x
.

(4) Let f : [1, ∞) → (−∞, 0] be a decreasing bijection (it follows that
f(1) = 0). Then f is order conjugate to 1 − x.

(5) Let f : (−∞, 0] → (0, 1] be an increasing bijection (it follows that
f(0) = 1). Then f is order conjugate to −1

x−1.

Proof. This Proposition easily follows from the following fact.
Let A, B ⊂ R be intervals such that either both f, g : A → B are decreasing

bijections or both are increasing bijections. Then there exist increasing bijec-
tions hf : A → A and hg : B → B such that f ◦ hf = hg ◦ g. This is because
we can always take an arbitrary hf and define hg = f ◦ hf ◦ g−1. □

Next we consider:

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K. Ali Akbar

Proposition 3.19. Let f : R → R be a continuous map with exactly two
non-ordinary points 0, 1 such that

(1) f(0) = 0.
(2) Either f|[1,∞) : [1, ∞) → [1, ∞) and it is an increasing bijection or

f|[1,∞) : [1, ∞) → (0, 1] and it is a decreasing bijection.
(3) Either f|(−∞,0] : (−∞, 0] → (−∞, 0] and it is an increasing bijection

or f|(−∞,0] : (−∞, 0] → [0, 1) and it is a decreasing bijection.
Then there are only 48 such maps up to order conjugacy.

Proof. Let f : R → R be a continuous map with exactly two non-ordinary
points 0, 1 such that it satisfies the three conditions mentioned in the state-
ment of Proposition 3.19. Then these non-ordinary points provides an ordered
partition {(−∞, 0), (0, 1), (1, ∞)} of R \ {0, 1}. Now by Proposition 2.19, Re-
mark 2.20, Proposition 3.10, Theorem 3.14, and Proposition 3.18, the map
f : R → R has exactly four choices in the interval (−∞, 0), exactly three
choices in the interval (0, 1), and exactly five choices in the interval (1, ∞) up
to order conjugacy. But because of the assumed conditions there are only 48
maps up to order conjugacy. See figure 1 (a). □

Remark 3.20. There are sixty eight maps (up to order conjugacy) having ex-
actly two non-ordinary points 0, 1 such that both are fixed.

Proof. By Corollary 2.18, if f : R → R be constant in a neighbourhood of a
point x0 then the end points of the maximal interval around x0 on which f is
constant, are non-ordinary. Hence by the arguments involved in the proof of
Proposition 3.19, this remark follows. See figure 1 (a). □

Now we have:

Proposition 3.21. Let f : R → R be a continuous map with exactly two
non-ordinary points 0, 1 such that

(1) f(1) = 1.
(2) Either f|[1,∞) : [1, ∞) → [1, ∞) and is an increasing bijection or

f|[1,∞) : [1, ∞) → (0, 1] and is a decreasing bijection.
(3) Either f|(−∞,0] : (−∞, 0] → (−∞, 0] and is a decreasing bijection or

f|(−∞,0] : (−∞, 0] → (0, 1] and is a increasing bijection.
Then there are only 8 such maps up to order conjugacy.

Proof. Proof follows from Theorem 3.14, Remark 2.20 and Propositions 3.10,
3.18 and 2.19 (see figure 1(b)). □

Remark 3.22. There are eleven maps on R (up to order conjugacy) having
exactly two non-ordinary points 0, 1 such that 1 is a fixed point and the image
of 0 is 1.

Proof. This remark follows from Corollary 2.18 together with the results used
for the proof of the Proposition 3.21 (see figure 1(b)). □

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The class of simple dynamical systems

Figure 1. Maps with exactly two non-ordinary points

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K. Ali Akbar

Next we have:

Proposition 3.23. Let f : R → R be a continuous map with exactly two
non-ordinary points 0, 1 such that

(1) f(0) = g(0) = 0.
(2) Either f|(−∞,0] : (−∞, 0] → (0, 1] and is a decreasing bijection or

f|(−∞,0] : (−∞, 0] → (−∞, 0] and is an increasing bijection.
(3) Either f|[1,∞) : [1, ∞) → (−∞, 0] and is a decreasing bijection or

f|[1,∞) : [1, ∞) → [0, 1) and is an increasing bijection.
Then there are only 8 such maps up to order conjugacy.

Proof. Proof follows from Theorem 3.14, Remark 2.20 and Propositions 3.10,
3.18 and 2.19 (see figure 1 (c)). □

Remark 3.24. There are eleven continuous maps on R (up to order conjugacy)
having exactly two non-ordinary points 0, 1 such that 0 is a fixed point and the
image of 1 is 0.

Proof. This remark follows from Corollary 2.18 together with results used for
the proof of the Proposition 3.23 (see figure 1 (c)). □

Now we are ready to consider our first main theorem:

Theorem 3.25 (Main Theorem 1). There are exactly 90 continuous maps on
R with exactly two non-ordinary points, up to order conjugacy.

Proof. Let a and b be the two non-ordinary points such that a < b. Then {a, b}
is invariant under f by Proposition 2.15. Hence at least one of these two points
is a fixed point; the other is either a fixed point or goes to a fixed point.

Case 1: Both a and b are fixed points.
Without loss of generality we can assume that a = 0 and b = 1. This is

because, let f be a continuous map having only two non-ordinary points a and
b. Let h : R → R be an increasing bijection such that h([a, b]) = [0, 1]. Then
consider g = hfh−1. Then g(0) = 0 and g(1) = 1. From Remark 3.20, it
follows that that there are only 68 continuous maps of this type up to order
conjugacy.

Case 2: f(a) = a = f(b).
Without loss of generality we can assume that a = 0, b = 1 (a similar proof

as in Case 1 will work). From Remark 3.24, it follows that there are only 11
maps of this type up to order conjugacy.

Case 3. f(b) = b = f(a).
Without loss of generality we can assume that a = 0 and b = 1 (a similar

proof as in Case 1 will work). From Remark 3.22, it follows that there are only
11 maps of this type up to order conjugacy.

Hence the proof. □

Remark 3.26. From Corollary 2.18, Remark 2.20, Theorem 3.14 and Proposi-
tions 3.10, 3.18 and 2.19, there are 16 somewhere constant continuous maps

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The class of simple dynamical systems

up to order conjugacy such that interval of constancy is bounded, 31 some-
where constant continuous maps up to order conjugacy such that the interval
of constancy is unbounded, 18 nowhere constant continuous maps up to order
conjugacy with unique critical point (among them 9 maps having unique criti-
cal point as a local maximum, remaining 9 maps having unique critical point as
a local minimum), 3 continuous maps up to order conjugacy with exactly two
critical points, and 22 continuous maps up to order conjugacy with no critical
points. Hence there are exactly 90 continuous maps (up to order conjugacy)
on R with having exactly two non-ordinary points. It gives another way of
counting involved in Theorem 3.25.

We can represent each member of this class as a digraph as follows.
Let f : R → R be continuous map having exactly two non-ordinary points.

Let I1, I2, I3 be non-singleton equivalence classes such that I1 < I2 < I3, and
define a graph (G, Vf ) with vertex set Vf = {I1, I2, I3}, and an edge from Ij
to Im if f(Ij) = Im, and a symbol I (respectively D) on this edge whenever
f is increasing (respectively decreasing) on Ij for j = 1, 2, 3. If there is a k-
cycle, label one of the symbols A, B, O in the least vertex (say J1) of the cycle
depends on the graph(fk|J1) is above the diagonal or below the diagonal or on
the diagonal respectively.

Note: If f is a decreasing homeomorphism then we can consider the labeled
digraph of f2 instead of the labeled digraph of f because of Proposition 3.4.

Example 3.27. Define f : R → R by

f(x) =

$
%

&

2x if x ≤ 0
x2 if 0 < x ≤ 1
2x − 1 if x > 1

The map f : R → R is continuous with exactly two non-ordinary points 0 and
1. The labeled digraph associated to the map f has three vertices I1, I2, I3 with
one loop at each vertex with an edge labeling I on each loop. The additional
symbol labeled to the vertices I1, I2 and I3 are B, A and A respectively. The
associated graph does not have any other edge.

Example 3.28.

f(x) =

$
%

&

−2x + 1 if x ≤ 0
1 if 0 < x ≤ 1
1
x

if x > 1

The map f : R → R is continuous with exactly two non-ordinary points 0
and 1. The labeled digraph associated to the map f : R → R has three vertices
I1, I2, I3 with a directed edge from I3 to I2 with edge labeling D. The vertex
I1 has an additional symbol labeled as A and the remaining vertices does not
have any other additional label. The associated graph does not have any other
edge.

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K. Ali Akbar

3.3. Class of continuous maps having finitely many non-ordinary points.

Let f : R → R be a continuous map from this class. Then the set of all non-
ordinary points is invariant by Proposition 2.15. By Corollary 2.18, the end
points of the maximal interval around every point on which f is constant are
non-ordinary. By Proposition 2.19, if x ∈ R is both critical and ordinary then
f is locally constant at x, and if x is ordinary then so is f(x) whenever f is
not locally constant in a neighbourhood of x. Also note that limx→−∞f(x)
and limx→+∞f(x) are non-ordinary whenever f has only finitely many non-
ordinary points. Hence the following informations will help us to characterizes
the set of all continuous maps from R to R having finitely many non-ordinary
points (see Theorem 3.32).

(1) graph(f) is above the diagonal or below the diagonal or on the diagonal
on each equivalence class (a, b) if f|(a,b) is increasing, and f(a) = a,
f(b) = b.
Related to this we will assign the symbols A, B, O on the vertex (a, b)
of the labeled digraph of (R, f) depends on the graph(f|(a,b)) is above
the diagonal or below the diagonal or on the diagonal respectively.

(2) Increasing or decreasing or constant on each equivalence class.
Related to this we will assign symbols I, D on the edge from the equiva-
lence class to itself of the labeled digraph of (R, f) depends on the graph
(f) on corresponding equivalence class is increasing or decreasing re-
spectively.

(3) If fk(I) = I, fm(I) ∕= I, m < k then consider fk|J and ask (1) for
least J ∈ {I, f(I), ..., fk−1(I)}
Related to this we will assign the symbols A, B, O on the vertex J of
the labeled digraph of (R, f) depends on the graph (fk|J) is above the
diagonal or below the diagonal or on the diagonal respectively.

Now we introduce some labeled digraph for each (R, f) as follows:
Let I1, I2, ..., In be non-singleton equivalence classes such that I1 < I2 <

... < In. Define a graph (G, Vf ) with vertex set Vf = {I1, I2, ..., In}, and
define an edge from Ij to Ik if f(Ij) = Ik. But this graph would not give
full information of the dynamical system (R, f). To achieve this, we give more
labels on each edge on the graph of the map, and on the least vertex of each
cycle(see (1), (2) and (3) for details). Observe that, if f, g are order conjugate
then the associated labeled digraph should be isomorphic.

Note: If f is a decreasing homeomorphism having odd number of non-
ordinary points then we can consider the graph of f2 instead of f because of
Proposition 3.4.

Image of each non-trivial equivalence Class
Let f : R → R be a continuous map having n − 1 non-ordinary points,

and I1 < I2 < ... < In be the n non-singleton equivalence classes of f. Then
f(I1) = Im for some m ∈ {1, 2, ..., n} or a constant; and f(I2) = Im+1 or a
constant if m = 1, and Im or Im+1 or a constant if m > 1. In general for

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The class of simple dynamical systems

3 ≤ k ≤ n − 2, f(Ik) = Il or Il−1 or Il+1 or a constant, l depends on m; and
f(In) = Ij or a constant; and f(In−1) = Ij−1 or a constant if j = n, and Ij−1
or Ij or a constant if j < n. Note that j depends on m.

Note: This information gives the possible choices of edge sets for the as-
signed labeled digraph.

Now we consider the following definitions:

Definition 3.29. A graph isomorphism between two graphs G and H can be
defined as a bijection f : VG → VH that such that a pair of vertices u, v is
adjacent in VG if and only if the image pair f(u), f(v) is adjacent in VH. In full
generality, a graph isomorphism f : G → H is a pair of bijections fV : VG → VH
and fE : EG → EH such that for every edge e ∈ EG, the endpoints of e are
mapped onto the endpoints of fE(e). A digraph isomorphism is an isomorphism
of the underlying graphs such that the edge correspondence preserves all edge
directions. A labeled digraph isomorphism is an isomorphism of the underlying
digraphs such that the correspondence preserves labeling. Two graphs are
isomorphic if there is an isomorphism from one to the other, or informally, if
their mathematical structures are identical.

Definition 3.30. Let (S, ≤S), (T, ≤T ) be two partially ordered sets. An order
isomorphism from (S, ≤S) to (T, ≤T ) is a surjective map h : S → T such that
for all u and v in S, h(u) ≤T h(v) if and only if u ≤S v. In this case, the posets
S and T are said to be order isomorphic. All surjective order isomorphisms are
bijective (see [3]).

Definition 3.31. Let f : R → R be a continuous map. A subset of R is said
to be a dynamically independent set if any two points of the set have disjoint
orbits. A subset of R is said to be maximal dynamically independent if it is
dynamically independent and no other super set is dynamically independent.

Now we are ready to prove our second main theorem.

Theorem 3.32 (Main Theorem 2). Let C be the class of all continuous self
maps of R, having finitely many non-ordinary points. Then

(1) for every member of C, there exists a maximal dynamically independent
set that is a finite union of intervals.

(2) two members of C are order conjugate if and only if they have order
isomorphic maximal dynamically independent set as in (1), and with
isomorphic labeled digraphs.

(3) every order isomorphism between such maximal dynamically indepen-
dent set as in (1) extends uniquely to an order conjugacy.

Proof. Let C be the class of all continuous self maps of R, having finitely many
non-ordinary points.

(1) Let f ∈ C and let z0 = x1 < x2 < ... < xn be the n non-ordinary
points of f. Then I1 = (−∞, x1), In+1 = (xn, ∞) and Ii = (xi, xi+1),
i = 1, ..., n−1 be the n+1 non-singleton equivalence classes by Lemma

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K. Ali Akbar

2.11. Choose yi ∈ Ii whenever f is not a constant on Ii for i = 1, 2, ..., n;
and let J be the set of all i such that yi has been chosen. Let z1 be
the least xi not in O(z0). Define, inductively, zi+1 as the least xi not
in O(zi). Let Z be the the set of all elements such that zi+1 /∈ O(zi)
(it may be empty but always finite). Define yi+1 = f

ki(yi) if ki is least
such that fki(Ii) = Ii for i ∈ J. Then consider M =

'
i∈J(yi, yi+1)∪Z.

Then M is a maximal dynamically independent set. Note that this M
is always non-empty.

(2) First part is easy because having maximal dynamically independent
set is invariant under order conjugacy. ie., if M1 is a maximal dynam-
ically independent set of f. Then h(M1) is a maximal dynamically
independent set of g whenever h is an order conjugacy from f to g.

Conversely, let f, g ∈ C have order isomorphic maximal dynamically
independent set as in (1) (say M1 and M2 respectively), and with
isomorphic labeled digraphs. Consider all non-empty intersection of
each equivalence classes of f with M1 and g with M2. Observe that
there is a one to one correspondence between these intersections. Be-
cause of maximal dynamical independency, we can extend restriction
of the order isomorphism on these sets to a homeomorphism on its cor-
responding equivalence class. By a similar proof as in Theorem 3.14 we
can extend it to a unique conjugacy from f to g since f and g have iso-
morphic labeled digraphs and order isomorphic maximal dynamically
independent sets.

(3) Easily follows from (2).

□

4. Summary

For n ∈ N, let Cn be the class of all continuous self maps of R with finitely
many non-ordinary points up to order conjugacy. For each n ∈ N, with the
help of Theorem 3.32, we can determine the cardinality of Cn. Note that the
cardinality of C1 is 26 and the cardinality of C2 is 90. Our main theorem
characterizes the class of all continuous self maps of R with finitely many non-
ordinary points up to order conjugacy.

Acknowledgements. The author is very thankful to the referee for giving
valuable suggestions. The author acknowledges SERB-MATRICS Grant No.
MTR/2018/000256 for financial support.

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The class of simple dynamical systems

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