The fixed-point property under induced interval maps of continua
Let f : I → I be a continuous map of a compact interval I and C(I) be the hyperspace of all compact subintervals of I equipped with the Hausdorff metric. We investigate the existence of the fixed-point property of subsets of C(I) with respect to any induced interval map F : C(I) → C(I). In particula...
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irk-123456789-1771352021-02-11T01:29:01Z The fixed-point property under induced interval maps of continua Robatian, D. Let f : I → I be a continuous map of a compact interval I and C(I) be the hyperspace of all compact subintervals of I equipped with the Hausdorff metric. We investigate the existence of the fixed-point property of subsets of C(I) with respect to any induced interval map F : C(I) → C(I). In particular, we prove that any nonempty subcontinuum of C(I) has the fixed-point property. Нехай f : I → I — неперервне вiдображення компактного iнтервалу I та C(I) — гiперпростiр усiх компактних пiдiнтервалiв I з метрикою Гаусдорфа. Вивчається властивiсть iснування нерухомої точки в пiдмножинах C(I) вiдносно iндукованого вiдображення F : C(I) → C(I). Зокрема, доведено, що будь-який непорожнiй пiдконтинуум C(I) має властивiсть iснування нерухомої точки. 2015 Article The fixed-point property under induced interval maps of continua / D. Robatian // Нелінійні коливання. — 2015. — Т. 18, № 1. — С. 102-111 — Бібліогр.: 10 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177135 517.9 en Нелінійні коливання Інститут математики НАН України |
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Let f : I → I be a continuous map of a compact interval I and C(I) be the hyperspace of all compact subintervals of I equipped with the Hausdorff metric. We investigate the existence of the fixed-point property of subsets of C(I) with respect to any induced interval map F : C(I) → C(I). In particular, we prove that any nonempty subcontinuum of C(I) has the fixed-point property. |
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Robatian, D. |
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Robatian, D. The fixed-point property under induced interval maps of continua Нелінійні коливання |
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Robatian, D. |
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Robatian, D. |
| title |
The fixed-point property under induced interval maps of continua |
| title_short |
The fixed-point property under induced interval maps of continua |
| title_full |
The fixed-point property under induced interval maps of continua |
| title_fullStr |
The fixed-point property under induced interval maps of continua |
| title_full_unstemmed |
The fixed-point property under induced interval maps of continua |
| title_sort |
fixed-point property under induced interval maps of continua |
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Інститут математики НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/177135 |
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The fixed-point property under induced interval maps of continua / D. Robatian // Нелінійні коливання. — 2015. — Т. 18, № 1. — С. 102-111 — Бібліогр.: 10 назв. — англ. |
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Нелінійні коливання |
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AT robatiand thefixedpointpropertyunderinducedintervalmapsofcontinua AT robatiand fixedpointpropertyunderinducedintervalmapsofcontinua |
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2025-07-15T15:09:35Z |
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2025-07-15T15:09:35Z |
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1837726094139064320 |
| fulltext |
UDC 517.9
THE FIXED-POINT PROPERTY UNDER INDUCED INTERVAL
MAPS OF CONTINUA
ВЛАСТИВIСТЬ IСНУВАННЯ НЕРУХОМОЇ ТОЧКИ
ДЛЯ IНДУКОВАНИХ ВIДОБРАЖЕНЬ ВIДРIЗКА НА КОНТИНУУМАХ
D. Robatian
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivska st., 3, Kyiv, 01601, Ukraine
Let f : I → I be a continuous map of a compact interval I and C(I) be the hyperspace of all compact subi-
ntervals of I equipped with the Hausdorff metric. We investigate the existence of the fixed-point property
of subsets of C(I) with respect to any induced interval map F : C(I) → C(I). In particular, we prove that
any nonempty subcontinuum of C(I) has the fixed-point property.
Нехай f : I → I — неперервне вiдображення компактного iнтервалу I та C(I) — гiперпростiр
усiх компактних пiдiнтервалiв I з метрикою Гаусдорфа. Вивчається властивiсть iснування не-
рухомої точки в пiдмножинах C(I) вiдносно iндукованого вiдображенняF : C(I) → C(I). Зокре-
ма, доведено, що будь-який непорожнiй пiдконтинуум C(I) має властивiсть iснування нерухомої
точки.
1. Introduction. Studying the fixed-point property for different classes of mathematical objects
has been attracting mathematicians’ attention since decades ago. Particularly, over the past half-
century the theory of fixed points revealed that it is an incredibly powerful tool in the study of
various phenomena, as the fundamental concepts of "stability" or "equilibrium"
might be described in terms of fixed points. That is why fixed-point techniques and theorems
have been frequently applied in such diverse fields of science ranging from biology, chemistry,
economics, engineering and physics to fields such as game theory and logic. In mathematics, and
particularly in topology, the problem of the "fixed-point property" (or briefly FPP) is of great
importance as well. A topological space X is said to have the fixed-point property if for any
continuous map f : X → X there exists x ∈ X such that f(x) = x. Among topological spaces,
Euclidean spaces were the first, for which the existence of the FPP was proved. This was done,
more than a century ago, in theorems which are known as Brouwer’s fixed-point theorems.
In the most general form, the Brouwer’s fixed-point theorems state that if A is a compact
convex subset of a topological space X, then every continuous map from A into itself has a fixed
point. Therefore, the simplest forms of Brouwer’s fixed-point theorems guarantee the existence
of at least one fixed-point in a compact interval or disk under any continuous self-map of that
interval or disk.
In this paper, we study the FPP in the hyperspace of all compact subintervals of a compact
interval I with respect to induced interval maps on it. Before moving further, let us introduce
the space we are going to consider in a more precise way.
Let f : X → X be a continuous map from a continuum X into itself. Denote by C(X)
the hyperspace of all compact connected subsets of X endowed with the Hausdorff metric. It
is known that if X is compact, so is the hyperspace C(X) (see [9, p. 52 – 63]). We define the
c© D. Robatian, 2015
102 ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 1
THE FIXED-POINT PROPERTY UNDER INDUCED INTERVAL MAPS . . . 103
induced continuous map F : C(X) → C(X) by setting F(xc) = {f(x) : x ∈ xc} for each
xc ∈ C(X). It is also known that continuity of f implies continuity of F . In the present work,
we consider the particular case where the continuum X equals the compact interval I = [0, 1].
More precisely, we assume that f is a continuous self-map of the interval I = [0, 1] and C(I) is
the hyperspace of all compact subintervals of I. Consequently, the induced map F : C(I) →
→ C(I) will be a map such thatF([a, b]) = {f(x) : x ∈ [a, b]}, for any a, b ∈ I. It was observed
that if X is a one-dimensional continuum, then there could exist a close connection between
the dynamics of the induced map F : C(I) → C(I) and those of the original inducing map
f : C(I) → C(I) (see [3, 4, 6, 8]). In particular, it was proved that if we consider the induced
map of the interval I, then the ω-limit set of a point of C(I) is either a union of singletons or a
finite subset of C(I). Nevertheless, the connection between the dynamics of an interval map f
and the induced map F on C(I) is not always that close. For example, the induced map never
happens to be transitive, even if f is transitive [1] (more generally, an induced map of graphs is
never transitive [7]).
The main question of this paper is then which characteristics a subset of C(I) must have, so
that the subset has the FPP. In order to do this, we will begin from the least complicated sets
and will develop the discussion to some of those subsets which, roughly speaking, have more
complex structures. Naturally, we will focus on the topological characteristics of these sets, since
the FPP is, in fact, a topological property. That is, two homeomorphic spaces either both have
or both lack the FPP (see [10]). Therefore, in Subsection 3.2, finite sets of C(I) are going to
be considered first, as they are apparently the simplest subsets of the space C(I). In there, we
will show that, in general case, these sets do not have the FPP. But it might also be observed
that if one adds a minimum number of certain conditions to these finite sets, then there occurs
a different story and the FPP holds for them. After that, in Subsection 3.3 we study a certain
class of infinite subsets of C(I) and prove that they do not necessarily include a fixed point.
Although, they always have a point of period two. Finally, in Subsection 3.4 we prove that any
nonempty subcontinuum of C(I) has the FPP. Before moving onto the main part of the paper,
we give some important definitions and explanations that are going to be used in the remainder
of the paper.
2. Preliminaries. In the remainder of the paper we always assume that
I equals the compact interval [0, 1] endowed with the usual Euclidean metric;
C(I) denotes the hyperspace of all compact connected subsets of I, endowed with the
Hausdorff metric;
the lowercase f denotes a continuous interval map from I into itself;
the uppercase F denotes the induced map of f from C(I) into itself, i.e., for any a, b ∈ I we
have that F([a, b]) = {f(x) : x ∈ [a, b]}.
We would like the reader to remember that, for the sake of saving time and space, in most of
the cases, especially when we state a new lemma or theorem, we introduceF without repeatedly
mentioning that F is the induced map of f. So, whenever we talk about F , we assume the
existence of f a priori.
In the forthcoming paragraphs, we would like to make some clarifying points regarding the
phase space C(I), as it helps us express the further argumentation more effectively on the one
hand and more accessible for the reader on the other.
Firstly, note that we treat each element of C(I), despite the set-nature they have, as a single
whole object. That is, if roughly speaking, any space consists of points, then each point of C(I)
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 1
104 D. ROBATIAN
y
x
a b
[a, b]c
ac
bc
L
0
1
1
Fig. 1. I = [0, 1] is plotted on the horizontal axis, the shaded triangle shows
I2y≥x and the dark black diagonal denoted by L shows the subset
of all degenerate pointervals of C(I). Points ac, [a, b]c, bc in I2y≥x are
related to a, [a, b], b in I, respectively.
is a nonempty closed interval in I. For this reason and in order to avoid any ambiguity, we
use the word pointerval to address an individual element of the space C(I). Moreover, it is
necessary to introduce an appropriate notation in order to remove any probable confusion.
Let a and b be two arbitrary points in I such that a ≤ b, then the notation J = [a, b] could
potentially lead to confusion, since it is not entirely clear whether we mean the ordinary interval
[a, b] = {x ∈ I|a ≤ x ≤ b} ⊆ I or we mean the pointerval [a, b] as a single element of C(I).
Therefore, to avert this misunderstanding, we use [a, b]c (or equivalently Jc) for a pointerval in
C(I) (i.e., a single point of the phase space C(I)) and [a, b] (equivalently J) to mean the set
{x ∈ I|a ≤ x ≤ b} ⊆ I. In addition, the pointerval [a, a]c, is called a degenerate pointerval and,
for ease, is denoted by ac. Alternatively, the phrase nondegenerate pointerval will be applied for
those pointervals that have distinct extremities in I, i.e., a 6= b. The phrase "single point" will
be used to refer to a single element of I.
Secondly, we will give a specific interpretation of the phase space C(I) since it provides
us with a very useful tool in studying the induced dynamics of C(I) (see [5], Example 5.1).
Precisely, we will build a homeomorphism between C(I) and the two dimensional space I2y≥x =
= {(x, y) ∈ I2 | y ≥ x}, that is, the subset of those points of the square I2 that are located on
and above the diagonal of I2 (the blue area in Fig. 1), so that one can interpret any pointerval of
C(I) as an ordinary point in I2y≥x. So, suppose that [a, b]c ∈ C(I) is an arbitrary pointerval. The
following homeomorphism is what we are looking for: [a, b]c ↔ (a, b), where (a, b) is a point
in I2y≥x, whose first and second coordinates are a and b respectively. Therefore, any pointerval
in C(I) is uniquely related to a point in I2y≥x and conversely. For example, if a is any point in
I , then the pointerval ac ∈ C(I) is related to a point on the lower vertex of the triangle I2y≥x
with respect to the mentioned homeomorphism. Conversely, any point of the lower vertex of
I2y≥x is related to a degenerate pointerval of C(I). In the remainder, the subset of all degenerate
pointervals of C(I) will be denoted by L, that is, L = {[a, a]c ∈ C(I)|a ∈ I}. Obviously, the
set L is invariant under any induced map F on C(I).
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 1
THE FIXED-POINT PROPERTY UNDER INDUCED INTERVAL MAPS . . . 105
y
x
(J0)c
Q4[(J0)c]
Q1[(J0)c]
Q2[(J0)c]
Q3[(J0)c]
0
1
1
Fig. 2. Q1[(J0)c],Q2[(J0)c], Q3[(J0)c] and Q4[(J0)c].
Here, it is worth mentioning the following point. We are completely aware that the terms
such as "intersection", "union", "disjointness", "inclusion" and etc. are naturally being used
for expressing relations among sets. Since according to the definition, a pointerval is a single
element of the phase space C(I), it does not make much sense to use the above mentioned
terms regarding the pointervals. Despite this fact and in order to avoid tediousness of writing
extra words, we will sometimes apply such terms for pointervals. We would like the reader to
remember that, in such cases, we use these terms for pointervals having in mind as if we regard
these pointervals as ordinary subintervals of I.
Let (J0)c be a pointerval in C(I). Then, we define (see Fig. 2)
Q1[(J0)c] = {Jc ∈ C(I)|min J0 < min J and max J0 < max J};
Q2[(J0)c] = {Jc ∈ C(I)|J0 ⊆ J};
Q3[(J0)c] = {Jc ∈ C(I)|min J0 > min J and max J0 > max J};
Q4[(J0)c] = {Jc ∈ C(I)|J0 ⊇ J}.
Obviously, we have thatQ1(Jc)∪Q2(Jc)∪Q3(Jc)∪Q4(Jc) = C(I), for any Jc ∈ C(I). Also,
Q2(Jc) ∩Q4(Jc) = {Jc}, while Jc belongs to neither of the disjoint sets Q1(Jc) and Q3(Jc).
Let N be a set of pointervals, i.e., N ⊆ C(I). Then, N∪ =
⋃
Jc∈N J and N∩ =
⋂
Jc∈N J.
Obviously, N∪ and N∩ are subsets of I. If N is strongly F-invariant, then N∪ and N∩ are
strongly f -invariant (although, N∩ may happen to be empty). The set N = {(Ji)c : i =
= 1, . . . , n;n ∈ N} is said to be a cycle under the action of F (or simply F-cycle), if poi-
ntervals (J1)c, (J2)c, . . . , (Jn)c produce a cycle under F , that is, F [(Ji)c] = (Ji+1)c, for all i =
= 1, 2, . . . , n− 1, and F [(Jn)c] = (J1)c. By card(N ) we denote the cardinality of the set N .
Let F : C(I) → C(I) be a continuous induced map. Then, we call a pointerval Jc ∈ C(I)
an eventually periodic pointerval (of period p ≥ 1) underF if there exists a number m ≥ 0 such
that Fm+p(Jc) = Fm(Jc). In the case where p = 1 we call Jc an eventually fixed pointerval.
Let S =
(
(Ji)c
)∞
i=1
be a sequence of pointervals, where Ji = [ai, bi] ⊆ I. Then, the
sequence S is said to be monotone if the sequences (ai)
∞
i=0 and (bi)
∞
i=0 are both monotone in
I. The sequence S is called bimonotone if both sequences S1 =
(
(J2k−1)c
)∞
k=1
and S2 =
=
(
(J2k)c
)∞
k=1
are monotone. Note that S1 and S2 can converge to different pointervals. So, a
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 1
106 D. ROBATIAN
monotone sequence is necessarily bimonotone, whilst the converse statement is not true (even
if S1 and S2 converge to the same pointerval. The monotone sequence S is called increasing
if (Ji+1)c ∈ Q2[(Ji)c] (or equivalently, Ji ⊆ Ji+1), for any i ≥ 1. Similarly, the monotone
sequence S is called decreasing if (Ji+1)c ∈ Q4[(Ji)c] (equivalently, Ji ⊇ Ji+1), for any i ≥ 1.
3. Existence of the FPP for subsets of C(I). In this section, we will examine some particular
subsets of C(I) to find out which of them have the FPP. In spite of the fact that connected
subsets of C(I) are seeming the most natural ones to be questioned, we will leave them until
the last part of this section. In the first stages, we begin with some of the simpler subsets that
are not necessarily connected. Namely, we are going to consider finite subsets of C(I) first, and
then we will develop the discussion for infinite subsets as well. Recall that for some of certain
finite subsets of the phase space I, it is extremely trivial to verify that the FPP does not hold.
For example, let A ⊂ I be a cycle of length n (≥ 2) under a continuous map f : I → I.
Then, obviously it does not have any fixed point. The same thing is true for finite subsets of
C(I). However, imposing some additional conditions to a finite set of pointervals can lead to
the existence of the FPP.
Before moving forward, let us state a few lemmas that are going to help us in the upcoming
discussion. Note that, for technical reasons, we have to frequently switch from the space C(I) to
I and conversely. That is why we often use F and f almost simultaneously in the same context
when we move from one space to the other. Although, in some cases one can apply either of
them without changing the concept.
3.1. Auxiliary lemmas.
Lemma 1. Let (J1)c ∈ C(I) be a nondegenerate pointerval and F : C(I) → C(I) be a
continuous induced map. Also, let (Ji)c = F i−1[(J1)c], for i = 1, 2, . . . . If there is a number
m ≥ 2 such that (Jm)c ∈ {Q2[(J1)c] ∪ Q4[(J1)c]} \ {(J1)c}, then there cannot exist any number
k ≥ 1 such that Fk[(Jm)c] = (J1)c.
Proof. Obviously, if (J2)c ∈ {Q2[(J1)c] ∪ Q4[(J1)c]} \ {(J1)c} (or equivalently J2 ⊂ J1 or
J2 ⊃ J1), then for any k ≥ 2 we have that (Jk)c ∈ {Q2[(J1)c] ∪Q4[(J1)c]} \ {(J1)c} and hence
(Jk)c 6= (J1)c.
Now, assume that m ≥ 3 be a number such that (Jm)c ∈ {Q2[(J1)c] ∪ Q4[(J1)c]} \ {(J1)c}
and (Ji)c ∈ C(I) \ {Q2[(J1)c] ∪Q4[(J1)c]}, for i = 2, 3, . . . ,m− 1. This implies that (Jm+k)c ∈
∈ {Q2[(Jk+1)c]∪Q4[(Jk+1)c]∪Q2[(J1)c]∪Q4[(J1)c]}\{(J1)c}, for any k ≥ 1, and this completes
the proof.
Lemma 2. Let (Ji)c, i = 1, 2, 3, be three nondegenerate pointervals in C(I) such that (J1)c 6=
6= (J2)c, (J1)c 6= (J3)c and J1∩J2∩J3 6= ∅. Also, letF : C(I) → C(I) be a continuous induced
map such that F [(Ji)c] = (Ji+1)c, for i = 1, 2. Then, there does not exist any number k ≥ 1, for
which holds Fk[(J3)c] = (J1)c.
Proof. Denote (Ji)c = F i−1[(J1)c], for all i ≥ 1. Let us consider all the possibilities that
can happen step by step.
The easiest case is when for any k ≥ 2 we have that (Jk)c is fixed. If this is the case, then
obviously there is nothing to prove. Hence, from this moment on we assume that this is not the
case.
First, if (J2)c ∈ {Q2[(J1)c]∪Q4[(J1)c]}\{(J1)c}, then obviously the desired statement holds
and there is nothing left to prove.
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 1
THE FIXED-POINT PROPERTY UNDER INDUCED INTERVAL MAPS . . . 107
So, assume that (J2)c ∈ Q1[(J1)c] or (J2)c ∈ Q3[(J1)c]. We discuss the first case thoroughly,
but will leave the second one to the reader as they are analogous.
Hence, suppose that (J2)c ∈ Q1[(J1)c]. At this moment again there can occur different
possibilities. If (J3)c ∈ Q2[(J2)c] ∪ Q4[(J2)c], then we are done. But if (J3)c ∈ {Q2[(J1)c] ∪
∪Q4[(J1)c]} \ {(J1)c}, then according to Lemma 1, (J3)c can never return to (J1)c under F-
iterations. In fact, Lemma 1 implies that if for any i ≥ 2 we have that (Ji)c ∈ Q2[(J1)c] ∪
∪Q4[(J1)c]∪Q2[(J2)c]∪Q4[(J2)c]∪ . . .∪Q2[(Ji−1)c]∪Q4[(Ji−1)c], then (J3)c cannot come back
to (J1)c underF . Therefore, we assume that this never happens. Assume that (J3)c ∈ Q1[(J2)c].
More precisely, we have that (J2)c ∈ Q1[(J1)c] and (J3)c ∈ Q1[(J2)c] (see Fig. 3(a)). In this case,
obviously, holds the inclusion J2 ⊂ J1 ∪ J3. But the last inclusion implies that J3 ⊂ J2 ∪ J4,
which, in turn, means that (Ji)c 6= (J1)c, for any i ≥ 4.
y
x
(a)
(J1)c
(J2)c
(J3)c
0
1
1
y
x
0
1
1
(b)
(J1)c
(J3)c
(J2)c
y
x
0
1
1
(c)
(J3)c
(J1)c
(J2)c
Fig. 3. Proof of Lemma 2.
Now, suppose that (J2)c ∈ Q1[(J1)c] and (J3)c ∈ Q1[(J1)c] ∩ Q3[(J2)c] (see Fig. 3(b)). One
can easily check that if this is the case, then holds J3 ⊂ J1 ∪ J2. But, this implies that Ji+2 ⊂
⊂ Ji∪Ji+1, for all i ≥ 1. In other words, for any i ≥ 4, we have that min J1 < min J3 < min Ji,
i.e., (Ji)c 6= (J1)c. Next, assume that (J2)c ∈ Q1[(J1)c] and (J3)c ∈ Q3[(J1)c] (see Fig. 3(c)). It
is easy to see that we have J1 ⊂ J2 ∪ J3. This implies that Ji ⊂ Ji+1 ∪ Ji+2, for all i ≥ 1. This
means that if i is an even number, then we have that max J1 < max J2 < max Ji, and if i is an
odd number, then min Ji < min J3 < min J1. So, obviously we have that (Ji)c 6= (J1)c, for any
i ≥ 4. Therefore, we discussed all the possible variants for the case where (J2)c ∈ Q1[(J1)c]
and, as we mentioned before, the second case (J2)c ∈ Q3[(J1)c] is left to the reader.
Lemma 2 is proved.
Lemma 3. Let N = {(Ji)c : i = 1, . . . , n;n ∈ N} ⊂ C(I) be a set of nondegenerate
pointervals such that N∪ is a connected subset of I. If N is a cycle with respect to a continuous
induced map F : C(I) → C(I), then card(N ) ≤ 2.
Proof. By way of contradiction, assume that n = card(N ) ≥ 3. It is clear that if N is a
cycle with respect to F , then the set N∪ is strongly invariant under f. This implies that there is
a single point x ∈ N∪, for which holds f(x) = x. Hence, there is at least one pointerval (Ji)c ∈
∈ N , i ∈ {1, . . . , n} such that x ∈ Ji. If this (Ji)c is the only pointerval of N that contains
x, then F [(Ji)c] = (Ji)c since x is an f -fixed point and this is obviously a contradiction as we
assumed that N is a cycle with at least three pointervals. So, there definitely exists another
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108 D. ROBATIAN
pointerval (Jj)c ∈ N , j ∈ {1, . . . , n}, i 6= j, that has the point x inside. Again and similarly
as above, we have that F [(Jj)c] 6= (Jj)c. In addition, F [Jj)c] 6= (Ji)c as they cannot produce a
cycle of period two. In the same way, one can conclude that x belongs to the all pointervals ofN .
In particular, we have that J1 ∩J2 ∩J3 6= ∅ and according to Lemma 2 there does not exist any
number k ≥ 1 such that Fk[(J3)c] = (J1)c. Therefore, the assumption n = card(N ) ≥ 3 we
made by contradiction is not true. On the other hand, it is primitive to check that if card(N ) ≤
2, then the set N can perfectly be a cycle.
Lemma 3 is proved.
Lemma 4. Let Ni = {(J i
1)c, (J
i
2)c, . . . , (J
i
ni
)c} ⊂ C(I), 1 ≤ i ≤ n, ni ∈ N, be sets of nonde-
generate pointervals such thatN∪i , for any i ∈ {1, . . . , n}, is a connected subset of I. Also, let for
any i, j ∈ {1, . . . , n}, i 6= j, holdsN∪i ∩N∪j = ∅. IfN =
⋃n
i=1Ni is a cycle under a continuous
induced map F : C(I) → C(I), then card(Ni) ≤ 2, for any i ∈ {1, . . . , n}.
Proof. It is not difficult to verify that ifN is a cycle under F , then all theNi’s have the same
cardinality and, in addition, each Ni is a cycle with respect to G = Fn. Hence, according to
Lemma 3, we have that card(Ni) ≤ 2, for any i ∈ {1, . . . , n}.
3.2. Finite subsets of C(I). Now, let us go back to the question mentioned at the beginning
of Section 3. As it was mentioned there, by adding some more conditions to a finite set of C(I)
we may obtain the FPP. By adding these conditions the range of finite sets we are going to
discuss will narrow. Answering the question relating the existence of FPP for all finite subsets
of C(I) needs further investigation. So, let us assume that N ⊂ C(I) is a nonempty, finite and
strongly F-invariant set such that N∩ 6= ∅. The question is then whether N always contains
a fixed, under F , pointerval or not. First, it is clear that each element of a finite and strongly
invariant set (no matter which phase space we are in) is periodic of period p (≥ 1). This fact
together with the following lemma are the key factors to answer the question.
Lemma 5. Let N = {(Ji)c : i = 1, . . . , n;n ∈ N} ⊂ C(I) be a nonempty set such that
N∩ 6= ∅. Also, let F : C(I) → C(I) be a continuous induced map.
(a) IfN is F-invariant, then each pointerval ofN is eventually fixed or eventually periodic of
period p = 2 with respect to F .
(b) IfN is stronglyF-invariant, then each pointerval ofN is fixed or periodic of period p = 2
with respect to F .
Proof. Suppose that (Ji)c is an arbitrary pointerval in N . As N is invariant with respect
to F and N∩ 6= ∅, Lemma 3 implies that if (Ji)c is a periodic pointerval of period p, then
p ≤ 2. Now, let us see what will happen if (Ji)c is not periodic at all. If (Ji)c never returns
to itself under F-iterations, and having in mind that N is F-invariant, then there definitely
exists a number m ≥ 1 such that Fm[(Ji)c] is fixed or of period two. Hence, until now we have
proved that ifN is only invariant, then any arbitrary pointerval (Ji)c ofN is eventually fixed or
eventually periodic of period two. But ifN is strongly invariant, then any pointerval (Ji)c ∈ N
has to be periodic. This together with what we have just shown above lead us to the fact that
(Ji)c is either fixed or of period 2.
Lemma 5 is proved.
Now, and according to Lemma 5, it is casy to see that the following theorem is true.
Theorem 1. Let N = {(Ji)c : i = 1, . . . , n;n ∈ N} ⊂ C(I) be a nonempty set such that
N∩ 6= ∅. Also, let F : C(I) → C(I) be a continuous induced map.
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THE FIXED-POINT PROPERTY UNDER INDUCED INTERVAL MAPS . . . 109
(a) If N is F-invariant, then it contains a periodic pointerval of period p ≤ 2 with respect
to F ;
(b) If N is strongly F-invariant and card(N ) is odd, then it necessarily contains a fixed
pointerval with respect to F .
3.3. Infinite subsets of C(I). Now in this part of the paper, we would like to discuss the
question regarding the FPP for infinite subsets of C(I). As in the previous part, we will add
similar conditions to infinite sets of C(I) and will focus only on this sort of sets.
More precisely, let O ⊂ C(I) be a nonempty, closed and infinite set such that O∩ 6= ∅.
Also, let O be an F-invariant set. Lemma 6 helps us characterize the trajectories of pointervals
that belong to O.
Lemma 6. Let (Ji)c, i = 1, 2, 3, be three pointervals in C(I) such that J1∩J2∩J3 6= ∅. Also,
let F : C(I) → C(I) be a continuous induced map such that F [(Ji)c] = (Ji+1)c, for i = 1, 2.
Then, there exists a number m ≥ 1 such that the sequence
(
(Ji)c
)∞
i=m
is bimonotonic.
Proof. Denote (Ji)c = F i−1[(J1)c], for all i ≥ 1. Suppose that (J1)c is any pointerval in
O. We know that O is F-invariant and, in addition, we have that O∩ 6= ∅. So, according to
Lemma 3, if (J1)c has a finite trajectory, then it has to be an eventually fixed or eventually
periodic of period p = 2.
Now, assume that (J1)c ∈ O is any pointerval that has an infinite trajectory. First, if there
is a number m ≥ 1 such that (Jm+1)c ∈ Q2[(Jm)c] ∪ Q4[(Jm)c], then obviously the sequence(
(Ji)c
)∞
i=m
is either increasing or decreasing and, hence, is bimonotone. So, assume that this
does not happen. Therefore, one of the following can occur:
(J2)c ∈ Q1[(J1)c] or (J2)c ∈ Q3[(J1)c]. As the argumentation for both of these cases are
analogous, similarly as in Lemma 2, we only discuss the first one and leave the second one to
the reader.
So, suppose that (J2)c ∈ Q1[(J1)c]. If (J3)c ∈ Q2[(J1)c] \ {(J1)c} or (J3)c ∈ Q4[(J1)c] \
{(J1)c}, then we have that (Ji+2)c ∈ Q2[(Ji)c] or (Ji+2)c ∈ Q4[(Ji)c], for all i ≥ 1, respectively.
And this clearly means that the sequence
(
(Ji)c
)∞
i=1
is bimonotone. As a matter of fact, this
can happen any time, that is, it is generally possible that there is a number m ≥ 1 such that
(Jm+2)c ∈ Q2[(Jm)c] \ {(Jm)c} or (Jm+2)c ∈ Q4[(Jm)c] \ {(Jm)c}, and in both cases we have
that the sequence
(
(Ji)c
)∞
i=m
is bimonotone.
Now, suppose that (J2)c ∈ Q1[(J1)c] and (J3)c ∈ Q1[(J2)c] (Fig. 3(a)). One can easily show
that in this case (and according to the assumption (Ji+1)c /∈ Q2[(Ji)c]∪Q4[(Ji)c], for any i ≥ 1,
that we made at the beginning of the proof), the only possible scenario that can take place
is that (Ji+1)c ∈ Q1[(Ji)c], for all i ≥ 1. And this means that the sequence
(
(Ji)c
)∞
i=1
is a
monotone and hence bimonotone sequence.
The next possibility is that (J2)c ∈ Q1[(J1)c] and (J3)c ∈ Q1[(J1)c] ∩ Q3[(J2)c] (Fig. 3(b)).
Also, we have that (Ji+2)c /∈ Q2[(Ji)c] ∪ Q4[(Ji)c] ∪ Q2[(Ji+1)c] ∪ Q4[(Ji+1)c], for any i ≥ 1.
If this is the case, then we have that J3 ⊂ J1 ∪ J2, which implies that Ji+2 ⊂ Ji ∪ Ji+1. And
this means that for any odd number i ≥ 1, we have that (Ji+2)c ∈ Q1[(Ji)c] and for any even
number i ≥ 2 we have that (Ji+2)c ∈ Q3[(Ji)c]. And this is exactly what we were looking for.
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110 D. ROBATIAN
The last possibility is when we have that (J2)c ∈ Q1[(J1)c] and (J3)c ∈ Q3[(J1)c] (Fig. 3(c)).
Exactly the same as in the previous cases, we assume that (Ji+2)c /∈ Q2[(Ji)c] ∪ Q4[(Ji)c] ∪
∪Q2[(Ji+1)c] ∪ Q4[(Ji+1)c], for any i ≥ 1. Clearly, we have that J1 ⊂ J2 ∪ J3, which in turn
implies that Ji ⊂ Ji+1 ∪ Ji+2, for all i ≥ 1. Hence, for any odd number i ≥ 1, we have that
(Ji+2)c ∈ Q3[(Ji)c], and for any even number i ≥ 2 we have that (Ji+2)c ∈ Q1[(Ji)c], i.e., the
sequence
(
(Ji)c
)∞
i=1
is bimonotone.
Lemma 6 is proved.
In fact, the next corollary immediately follows from Lemma 6.
Corollary 1. Let O ⊂ C(I) be a nonempty, closed and infinite set such that O∩ 6= ∅. Also,
let F : C(I) → C(I) be a continuous induced map. If O is invariant with respect to F , then for
any pointerval Jc ∈ O, holds either of the following:
(a) the pointerval Jc is eventually periodic of period p ≤ 2;
(b) the trajectory of Jc is infinite and there is a number m ≥ 1 such that the sequence(
F i(Jc)
)∞
i=m
is bimonotone.
Note that if a sequence ((Ji)c)
∞
i=1 is bimonotone, then the subsequences ((J2k−1)c)∞k=1 and
((J2k)c)
∞
k=1 not necessarily converge to the same pointerval. And as it is possible that these
subsequences converge to two different pointervals, hence the described set O in Corollary 1
does not necessarily contain an F-fixed pointerval inside. On the other hand, it definitely con-
tains a pointerval of period two. Therefore, the next theorem holds:
Theorem 2. Let O ⊂ C(I) be a nonempty, closed and infinite set such that O∩ 6= ∅. Also, let
F : C(I) → C(I) be a continuous induced map. If O is invariant with respect to F , then there
exists a pointerval Jc ∈ O with F2(Jc) = Jc.
3.4. Subcontinuums of C(I). In the remainder, we prove that any nonempty subcontinuum
of C(I) has the FPP.
Theorem 3. Let C be a nonempty continuum of C(I) and F : C(I) → C(I) be a continuous
induced map. If C is invariant with respect to F , then C has the FPP.
Proof. Suppose that (J0)c is an arbitrary pointerval in C. It is obvious that holds at least one
of the following inclusions:
(a) F [(J0)c] ∈ Q1[(J0)c] ∩ C,
(b) F [(J0)c] ∈ Q2[(J0)c] ∩ C,
(c) F [(J0)c] ∈ Q3[(J0)c] ∩ C,
(d) F [(J0)c] ∈ Q4[(J0)c] ∩ C.
Note that we said "at least"one the above mentioned items holds, because clearly in general
case (b) and (d) can happen simultaneously. But if this is the case, then (J0)c is a fixed pointerval
under F and there is nothing left to prove. So, suppose that for any pointerval of C occurs
exactly one of the above inclusions. Let C1 = {Jc ∈ C : F(Jc) ∈ Q1(Jc)}, C2 = {Jc ∈ C :
F(Jc) ∈ Q2(Jc)}, C3 = {Jc ∈ C : F(Jc) ∈ Q3(Jc)} and C4 = {Jc ∈ C : F(Jc) ∈ Q4(Jc)}.
Clearly, we have that C = C1 ∪ C2 ∪ C3 ∪ C4. First, we show that C2 and C4 cannot be empty
simultaneously, that is, C2 ∪ C4 6= ∅.
Seeking a contradiction, suppose that C2∪C4 = ∅. Consequently, we obtain that C = C1∪C3.
But this is a contradiction as the nonempty connected set C cannot be expressed as a union of
two open (with respect to the induced topology on C) and pairwise disjoint sets C1 and C3. In
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THE FIXED-POINT PROPERTY UNDER INDUCED INTERVAL MAPS . . . 111
order to understand why C1 is open, take an arbitrary pointerval (J0)c in C1. Then, it is not hard
to verify that there always exists an ε > 0 such that Bε[(J0)c] ⊂ C1, where Bε[(J0)c] = {Jc ∈
∈ C : distH[Jc, (J0)c] < ε}. This means that (J0)c is an interior point for C1 and from here
implies that C1 is open in C. Similarly, one can prove that C3 is open as well. Until now, we proved
that the assumption we made by contradiction is not true and hence we have that C2 ∪ C4 6= ∅.
Now, we will show that the fact that not both C2 and C4 are empty at the same time guarantees
the existence of a fixed pointerval in C. With no loss of generality, assume that C2 6= ∅. Also,
suppose that (J1)c be any pointerval in C2. We have that F i+1[(J1)c] ∈ Q2
[
F i[(J1)c]
]
, for any
i ≥ 0. Hence,
(
F i[(J1)c]
)∞
i=0
is an increasing sequence of pointervals in the bounded set C. So,(
F i[(J1)c]
)∞
i=0
is convergent to a fixed pointerval in C.
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Received 11.09.14
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