On the Set of Fixed Points and Periodic Points of Continuously Differentiable Functions

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Volume 2013, Article ID 929475,5pages http://dx.doi.org/10.1155/2013/929475

Research Article

On the Set of Fixed Points and Periodic Points of

Continuously Differentiable Functions

Aliasghar Alikhani-Koopaei

Department of Mathematics, Berks College, Pennsylvania State University, Tulpehocken Road, P.O. Box 7009, Reading, PA 19610-6009, USA

Correspondence should be addressed to Aliasghar Alikhani-Koopaei; [email protected]

Received 6 December 2012; Accepted 16 February 2013

Academic Editor: Paolo Ricci

Copyright © 2013 Aliasghar Alikhani-Koopaei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In recent years, researchers have studied the size of different sets related to the dynamics of self-maps of an interval. In this note we investigate the sets of fixed points and periodic points of continuously differentiable functions and show that typically such functions have a finite set of fixed points and a countable set of periodic points.

1. Introduction and Notation

The set of periodic points of self-maps of intervals has been studied for different reasons. The functions with smaller sets of periodic points are more likely not to share a periodic point. Of course, one has to decide what “big” or “small” means and how to describe this notion. In this direction one would be interested in studying the size of the sets of periodic points of self-maps of an interval, in particular, and other

sets arising in dynamical systems in general (see [1–5]). For

example, typically continuous functions have a first category

set of periodic points (see [1,5]). This result was generalized

in [2] for the set of chain recurrent points. At times, even

the smallness of these sets in some sense could be useful. For

example, in [6] we showed that two commuting continuous

self-maps of an interval share a periodic point if one has a

countable set of periodic points. Schwartz (see [7]) was able to

show that if one of the two commuting continuous functions is also continuously differentiable, then it would necessarily follow that the functions share a periodic point. Schwartz’s

result along with the results given in [6] may suggest that

continuously differentiable functions have a countable set of periodic points. This is not true in general. However, in this note we show that typically such functions have a finite set of fixed points and a countable set of periodic points. Here

𝐹1(𝑓) = {𝑥 : 𝑓(𝑥) = 𝑥}denotes the set of fixed points of𝑓.

For𝑓and𝑛 ∈Nwe define𝑓𝑛(𝑥)by induction:

𝑓𝑛(𝑥) = 𝑓 (𝑓𝑛−1(𝑥))

with𝑓0(𝑥) = 𝑥, 𝑓1(𝑥) = 𝑓 (𝑥) .

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The orbit of𝑥under𝑓is given by the sequence orb(𝑥, 𝑓) =

{𝑓𝑖_(𝑥)}∞

𝑖=0. For𝑛 ∈N, let𝐹𝑛(𝑓) = {𝑥 : 𝑓𝑛(𝑥) = 𝑥}and𝑃𝑓𝑛be

the set of periodic points of order𝑛; that is,

𝑃_𝑓𝑛= 𝐹_𝑛(𝑓) \ ∪_𝑘<𝑛𝐹_𝑘(𝑓) . (2)

Two functions on a given interval are said to be of the same monotone type if both are either strictly increasing or strictly

decreasing on that interval. Here, for a partition 𝑇of the

interval[𝑎, 𝑏],‖𝑇‖is the length of the largest subinterval of

𝑇,𝐵(𝑓, 𝜖)is the open ball about𝑓with radius𝜖,𝐴∘denotes

the set of interior points of𝐴, and𝜆(𝐼)is the length of the

interval𝐼.

2. Continuously Differentiable Functions

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from𝐼into itself, respectively. Recall that the usual metrics𝜌₀ and𝜌₁on𝐶(𝐼, 𝐼)and𝐶1(𝐼, 𝐼), respectively, are given by

𝜌₀(𝑓, 𝑔) =sup

𝑥∈𝐼󵄨󵄨󵄨󵄨𝑓(𝑥) − 𝑔(𝑥)󵄨󵄨󵄨󵄨 for𝑓, 𝑔 ∈ 𝐶 (𝐼, 𝐼) ,

𝜌₁(𝑓, 𝑔) = 𝜌₀(𝑓, 𝑔) + 𝜌₀(𝑓󸀠_{, 𝑔}󸀠₎ _for_{𝑓, 𝑔 ∈ 𝐶}1_{(𝐼, 𝐼) .} (3)

It is well known that the metric spaces (𝐶(𝐼, 𝐼); 𝜌₀)and

(𝐶1_{(𝐼, 𝐼); 𝜌}

1)are complete and hence Baire’s category theorem

holds in these spaces. We say that a typical function in𝐶(𝐼, 𝐼)

or𝐶1(𝐼, 𝐼)has a certain property if the set of those functions

which does not have this property is of first category in𝐶(𝐼, 𝐼)

or in𝐶1(𝐼, 𝐼). (Some authors prefer using the term generic

instead of typical.) It is known that typically continuous

self-maps of an interval have 𝜎-perfect, measure zero sets

of periodic points (see [2]). Here we show that typically

members of(𝐶1(𝐼, 𝐼), 𝜌₁)have a finite set of fixed points and

a countable set of periodic points.

Lemma 1. Let𝑛 ≥ 1and𝑓 ∈ 𝐶1(𝐼, 𝐼)so that𝐹_𝑛(𝑓)is not

finite. Then there exists 𝑥₀ ∈ 𝐼 so that𝑓𝑛(𝑥₀) = 𝑥₀ and

(𝑓𝑛₎󸀠_(𝑥 0) = 1.

Proof. Suppose𝐹_𝑛(𝑓)is not finite, and then we can choose

a strictly monotone sequence in 𝐹_𝑛(𝑓). Without loss of

generality, assume {𝑥_𝑖}∞_𝑖=1 ⊂ 𝐹_𝑛(𝑓)so that 𝑥_𝑖 < 𝑥_𝑖+1 for

each 𝑖 ≥ 1. Let ℎ(𝑥) = 𝑓𝑛(𝑥) − 𝑥, and then, for each 𝑖,

ℎ(𝑥_𝑖) = ℎ(𝑥_𝑖+1) = 0. Thus by Rolle’s Theorem we have𝑦_𝑖,

𝑥_𝑖 < 𝑦_𝑖 < 𝑥_𝑖+1so thatℎ󸀠(𝑦_𝑖) = 0. Let𝑥₀ = lim_{𝑖 → ∞}𝑥_𝑖, then

𝑥₀ ∈ 𝐼, and lim_{𝑖 → ∞}𝑦_𝑖 = 𝑥₀, and from the continuity of𝑓𝑛

and(𝑓𝑛)󸀠it follows that𝑓𝑛(𝑥₀) = 𝑥₀and(𝑓𝑛)󸀠(𝑥₀) = 1.

Lemma 2. For each𝑓 ∈ 𝐶1(𝐼, 𝐼)and0 < 𝜖 < 1there is a

polynomial𝑝 : 𝐼 → 𝐼𝑜with𝜌₁(𝑓, 𝑝) < 𝜖.

Proof. Let𝑀 = sup_𝑥∈𝐼{|𝑓󸀠(𝑥)|, |𝑎|, |𝑏|, 1}where𝐼 = [𝑎, 𝑏].

Take𝛽 = 𝜖/8𝑀,𝛼 = 𝛽(𝑎 + 𝑏)/2, and𝑔(𝑥) = 𝛼 + (1 − 𝛽)𝑓(𝑥).

It is easy to see that0 < 𝛽 < 1,𝑔(𝐼) ⊂ 𝐼𝑜, and𝜌₁(𝑓, 𝑔) < 𝜖/4.

Let

𝜖1= 1₂_{𝑥∈[𝑎,𝑏]}inf {𝑔 (𝑥) − 𝑎, 𝑏 − 𝑔 (𝑥) ,_{8 (𝑏 − 𝑎)}𝜖 ,₈𝜖} (4)

and𝑠(𝑡)be a polynomial with𝜌₀(𝑔󸀠, 𝑠) < 𝜖₁/(𝑏 − 𝑎 + 1). Then

𝑝(𝑥) = 𝑔(𝑎) + ∫_𝑎𝑥𝑠(𝑡)𝑑𝑡is a polynomial and𝑔(𝑥) − 𝑝(𝑥) =

∫_𝑎𝑥(𝑔󸀠_{(𝑡) − 𝑠(𝑡))𝑑𝑡}_{. Thus we have}

𝜌₁(𝑔, 𝑝) = 𝜌₀(𝑔󸀠, 𝑠) + 𝜌₀(𝑔, 𝑝) ≤ 𝜖1

(𝑏 − 𝑎) + 1+

𝜖₁(𝑏 − 𝑎) (𝑏 − 𝑎) + 1

= 𝜖1< 𝜖₄,

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hence𝜌₁(𝑓, 𝑝) ≤ 𝜌₁(𝑓, 𝑔) + 𝜌₁(𝑔, 𝑝) < 𝜖/2. From𝑔(𝐼) ⊂ [𝑎 +

2𝜖₁, 𝑏 − 2𝜖₁]and𝜌₁(𝑔, 𝑝) ≤ 𝜖₁it follows that𝑝(𝐼) ⊂ 𝐼𝑜.

Lemma 3. Let𝑛 > 2be a positive integer and𝑓 ∈ 𝐶1(𝐼, 𝐼).

If𝑥₀is a periodic point of order 𝑛with(𝑓𝑛)󸀠(𝑥₀) = 1, then

orb(𝑥₀, 𝑓) ∩ {𝑎, 𝑏} = 0.

Proof. We show that𝑎 ∉orb(𝑥₀, 𝑓). The case𝑏 ∉ orb(𝑥₀, 𝑓)

follows similarly. Let orb(𝑥₀, 𝑓) = {𝑎 = 𝑥₀, 𝑥₁, . . . , 𝑥_𝑛 = 𝑥₀}

with𝑥_𝑖 = 𝑓(𝑥_𝑖−1)for1 ≤ 𝑖 ≤ 𝑛. Since𝑛 > 2, there exist

𝑥_𝑖 ∈ orb(𝑥₀, 𝑓) ∩ (𝑎, 𝑏)and1 ≤ 𝑠 ≤ 𝑛 − 1so that𝑓𝑠(𝑥_𝑖) =

𝑎. Since𝑓𝑠 ∈ 𝐶1(𝐼, 𝐼)and(𝑓𝑠)󸀠(𝑥₀) ̸= 0, there exists𝛿₁ > 0

so that𝑎 < 𝑥_𝑖− 𝛿₁ < 𝑥_𝑖 < 𝑥_𝑖+ 𝛿₁ < 𝑏and 𝑓𝑠 is strictly

increasing or strictly decreasing on𝐽 = [𝑥_𝑖− 𝛿₁, 𝑥_𝑖+ 𝛿₁]. Let

𝑓𝑠_{be strictly increasing on}_𝐽_{, and then for}_𝑥

𝑖−𝛿1≤ 𝑥 ≤ 𝑥𝑖we

have𝑓𝑠(𝑥) ≤ 𝑓𝑠(𝑥_𝑖) = 𝑎; hence,𝑓𝑠|_[𝑥_𝑖_−𝛿₁_,𝑥_𝑖_]= 𝑎, implying that

(𝑓𝑠₎󸀠_(𝑥

𝑖) = 0, a contradiction to(𝑓𝑛)󸀠(𝑥0) = 1. On the other

hand when𝑓𝑠is strictly decreasing on𝐽, for𝑥_𝑖≤ 𝑥 ≤ 𝑥_𝑖+ 𝛿₁

we have𝑓𝑠(𝑥) ≤ 𝑓𝑠(𝑥_𝑖) = 𝑎; hence,𝑓𝑠|_[𝑥_𝑖_,𝑥_𝑖_+𝛿₁_]= 𝑎implying

that(𝑓𝑠)󸀠(𝑥_𝑖) = 0, a contradiction to(𝑓𝑛)󸀠(𝑥₀) = 1.

Lemma 4. For𝑘 ≥ 1, the set

E𝑘= {𝑓 ∈ 𝐶1(𝐼, 𝐼) : 𝐹𝑘(𝑓) ∩ {𝑥 : (𝑓𝑘)󸀠(𝑥) = 1} ̸= 0}

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is closed in𝐶1(𝐼, 𝐼).

Proof. Let𝑔_𝑛 ∈ E_𝑘 and 𝜌₁(𝑔_𝑛, 𝑔) → 0. Then there exists

{𝑡_𝑛}∞_𝑛=1 ⊂ 𝐼such that𝑔𝑘_𝑛(𝑡_𝑛) = 𝑡_𝑛and(𝑔_𝑛𝑘)󸀠(𝑡_𝑛) = 1. Without

loss of generality, we may assume that lim_{𝑛 → ∞}𝑡_𝑛 = 𝑡₀, then

𝑡₀∈ 𝐼. Let𝜖 > 0be arbitrary. Due to the uniform continuity of

𝑔𝑘_on_𝐼_{there exists a positive integer}_𝑛

1such that, for𝑛 ≥ 𝑛1,

|𝑔𝑘(𝑡𝑛) − 𝑔𝑘(𝑡0)| < 𝜖. Since 𝜌0(𝑔𝑛, 𝑔) → 0, there exists a

positive integer𝑛₂ such that, for 𝑛 ≥ 𝑛₂,𝜌₀(𝑔_𝑛𝑘, 𝑔𝑘) < 𝜖.

Choose the positive integer𝑛₃so that|𝑡_𝑛− 𝑡₀| < 𝜖for𝑛 ≥ 𝑛₃,

and let𝑚₁=max(𝑛₁, 𝑛₂, 𝑛₃). Then for𝑚 ≥ 𝑚₁we have

󵄨󵄨󵄨󵄨

󵄨𝑔𝑘(𝑡0) − 𝑡0󵄨󵄨󵄨󵄨_󵄨

≤ 󵄨󵄨󵄨󵄨󵄨𝑔𝑘(𝑡₀) − 𝑔𝑘(𝑡_𝑚_{)󵄨󵄨󵄨󵄨󵄨 +}_󵄨𝑔󵄨󵄨󵄨󵄨 𝑘(𝑡_𝑚) − 𝑔_𝑚𝑘 (𝑡_𝑚_{)󵄨󵄨󵄨󵄨󵄨} + 󵄨󵄨󵄨󵄨󵄨𝑔𝑘𝑚(𝑡𝑚) − 𝑡0󵄨󵄨󵄨󵄨_󵄨

≤ 𝜖 + 𝜌₀(𝑔𝑘, 𝑔𝑘_𝑚_{) + 󵄨󵄨󵄨󵄨𝑡}_𝑚− 𝑡₀󵄨󵄨󵄨󵄨 ≤ 3𝜖,

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Implying that𝑔𝑘(𝑡₀) = 𝑡₀. We also have

󵄨󵄨󵄨󵄨 󵄨󵄨(𝑔𝑘𝑛)

󸀠

(𝑡𝑛) − (𝑔𝑘)󸀠(𝑡𝑛)󵄨󵄨󵄨󵄨󵄨󵄨 ≤_{𝑛 → ∞}lim𝜌1(𝑔𝑛𝑘, 𝑔𝑘) = 0. (8)

Thus we have lim_{𝑛 → ∞}|1 − (𝑔𝑘)󸀠(𝑡_𝑛)| = |1 − (𝑔𝑘)󸀠(𝑡₀)| = 0and

𝑔 ∈E_𝑘.

Theorem 5. There exists a residual subset𝑀₁of𝐶1(𝐼, 𝐼)such

that, for every𝑓 ∈ 𝑀₁,𝐹₁(𝑓)is finite.

Proof. If 𝐹₁(𝑓) has infinitely many elements, then from

Lemma1it follows that there exists a point𝑥₀ ∈ 𝐹₁(𝑓)so

that𝑓󸀠(𝑥₀) = 1. Let

E₁= {𝑓 ∈ 𝐶1(𝐼, 𝐼) : ∃𝑥0∈ 𝐼 ∩ 𝐹1(𝑓) , 𝑓󸀠(𝑥0) = 1} . (9)

From Lemma4the setE₁is closed in𝐶1(𝐼, 𝐼). To show that

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Lemma2, there exists a polynomial𝑝such that𝜌₁(𝑓, 𝑝) < 𝜖/2 and𝑝(𝐼) ⊂ [𝑎 + 𝜖₁, 𝑏 − 𝜖₁]for some0 < 𝜖₁< 𝜖/2. Let

𝑀 (𝑝 ) = {𝑥 : 𝑝󸀠(𝑥) = 1} = {𝑡₁, 𝑡₂, . . . , 𝑡_𝑚} . (10)

Choose0 < 𝜖₂< 𝜖₁so that𝑝(𝑥) − 𝑥 ̸= 𝜖₂for all𝑥 ∈ 𝑀(𝑝 )and

takeℎ = 𝑝 − 𝜖₂. Thenℎ ∈ 𝐶1(𝐼, 𝐼),𝜌₁(ℎ, 𝑓) < 𝜖,𝑀(ℎ) = {𝑥 :

ℎ󸀠_{(𝑥) = 1} = 𝑀(𝑝)}_{, and}_{ℎ(𝑥) ̸= 𝑥}_on_𝑀(ℎ)_{, implying that}_E 1

is of first category and𝑀₁= 𝐶1(𝐼, 𝐼) \E₁is residual.

Theorem 6. The set of functions𝑓 ∈ 𝐶1(𝐼, 𝐼)with 𝑓(𝑎) =

𝑏, 𝑓(𝑏) = 𝑎, and(𝑓2)󸀠(𝑎) = 1is a nowhere dense subset of

𝐶1_{(𝐼, 𝐼)}_.

Proof. Let 𝐴 = {𝑓 ∈ 𝐶1(𝐼, 𝐼) : 𝑓(𝑎) = 𝑏, 𝑓(𝑏) =

𝑎, and (𝑓2)󸀠(𝑎) = 1}. It is easy to see that𝐴is a closed subset of𝐶1(𝐼, 𝐼). To show that𝐴is nowhere dense, let𝑓 ∈ 𝐶1(𝐼, 𝐼) and𝜖 > 0. Choose0 < 𝛾 < min{𝜖/(4(𝑏 − 𝑎) + 4), 1}and

𝑎 < 𝑐 < 𝑏such that|𝑓(𝑡) − 𝑓(𝑎)| < (𝑏 − 𝑎)[1 − 𝜖/4] for

𝑡 ∈ [𝑎, 𝑐]. Letℎ(𝑥) = 𝑓(𝑥) + ∫_𝑎𝑥𝑘(𝑡)𝑑𝑡, where

𝑘 (𝑡) = { { { { { { { { { { { { { { { { { { { { { { {

−𝛾, 𝑡 = 𝑎,

0, 𝑡 = 𝑎 +𝑐 − 𝑎 4 , 𝛾

3, 𝑡 = 𝑎 + (𝑐 − 𝑎)2 , 0, 𝑡 = 𝑐, 𝑡 = 𝑏,

linear, otherwise.

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It is clear thatℎ(𝑎) = 𝑓(𝑎) = 𝑏,ℎ(𝑏) = 𝑓(𝑏) = 𝑎,ℎ󸀠(𝑎) =

𝑓󸀠_{(𝑎) − 𝛾}_{, and}_ℎ󸀠_{(𝑏) = 𝑓}󸀠_(𝑏)_{. It is easy to see that}_{ℎ ∈ 𝐶}1_{(𝐼, 𝐼)}_,

(ℎ2₎󸀠_{(𝑎) = (ℎ}2₎󸀠_{(𝑏) ̸= 1}_{, and}_𝜌

1(𝑓, ℎ) < 𝜖/2.

Theorem 7. Let𝜖 > 0,𝑓 ∈ 𝐶1(𝐼, 𝐼), and𝐴_𝑛 = {𝑥 : (𝑓𝑛)󸀠(𝑥) =

1} ∩ 𝐹_𝑛(𝑓)be a finite set, and𝑜𝑟𝑏(𝑥, 𝑓) ⊂ (𝑎, 𝑏)for𝑥 ∈ 𝐴_𝑛.

Then there exists a function𝑔 ∈ 𝐶1(𝐼, 𝐼)with𝜌₁(𝑓, 𝑔) < 𝜖/4

such that{𝑥 : (𝑔𝑛)󸀠(𝑥) = 1} ∩ 𝐹_𝑛(𝑔) = 0.

Proof. Let𝐴_𝑛 = {𝑥 : (𝑓𝑛)󸀠(𝑥) = 1} ∩ 𝐹_𝑛(𝑓) = {𝑧_𝑗}𝑙_𝑗=1. Let

𝐵 = {𝑡𝑗}𝑘𝑗=1be the elements of𝐴𝑛with distinct orbits, that is,

orb(𝑡_𝑖, 𝑓) ∩orb(𝑡_𝑗, 𝑓) = 0for𝑖 ̸= 𝑗and∪𝑘_𝑗=1orb(𝑡_𝑗, 𝑓) ⊇ 𝐴_𝑛.

It is clear that each𝑡_𝑗 ∈ 𝐵is a periodic point of𝑓with some

period𝑚where𝑚is a factor of𝑛. This suggests that if one

can construct a function𝑔that is sufficiently close to𝑓, and

either𝑡_𝑗∉ 𝑃_𝑓𝑚or(𝑓𝑚)󸀠(𝑡_𝑗) ̸= 1, then

{𝑥 : (𝑔𝑛)󸀠(𝑥) = 1} ∩ 𝐹_𝑛(𝑔) ⊆ 𝐴_𝑛\orb(𝑡_𝑗, 𝑓) . (12)

By repeating this process for each𝑡_𝑗 ∈ 𝐵in a finite number

of steps we can construct the desired function𝑔. Thus for

convenience, we assume that, for1 ≤ 𝑗 ≤ 𝑘,𝑧_𝑗 ∈ 𝑃_𝑓𝑛.

Consider the partition𝑇of [𝑎, 𝑏]obtained by {𝑓𝑖(𝑡_𝑗), 1 ≤

𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑘},𝜖₁ = (1/4)min{𝜖, ‖ 𝑇 ‖}, and choose a

positive𝛿less than𝜖₁such that, for each𝑖, 1 ≤ 𝑖 ≤ 𝑛and each

𝑗, 1 ≤ 𝑗 ≤ 𝑘, and if|𝑡−𝑡_𝑗| < 𝛿, then|𝑓𝑖(𝑡)−𝑓𝑖(𝑡_𝑗)| < 𝜖₁. Given

that, for𝑥 ∈ 𝐴_𝑛, orb(𝑥, 𝑓) ⊂ (𝑎, 𝑏),(𝑓𝑛)󸀠is continuous, and

(𝑓𝑛)󸀠(𝑥) = 1on𝐵, we may choose the nondegenerate closed

intervals𝐻_𝑗, 1 ≤ 𝑗 ≤ 𝑘of length less than𝛿and the positive

numbers𝛾_𝑗such that

(i)𝑡_𝑗 ∈ (𝐻_𝑗)∘ is the midpoint of 𝐻_𝑗 and 𝑓𝑖(𝐻_𝑗) ∩

orb(𝑡_𝑗, 𝑓) = {𝑓𝑖(𝑡_𝑗)}for0 ≤ 𝑖 ≤ 𝑛,

(ii)𝑓is strictly monotone on each𝐻_𝑗,𝑖 = 𝑓𝑖(𝐻_𝑗)for0 ≤

𝑖 ≤ 𝑛 − 1,1 ≤ 𝑗 ≤ 𝑘, where𝐻_𝑗,0= 𝐻_𝑗,

(iii) the intervals𝐻_𝑗,𝑖 = 𝑓𝑖(𝐻_𝑗)are mutually disjoint for

1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑘,

(iv)𝑎 <min(𝐻_𝑗,𝑖) − 𝛾_𝑗𝜆(𝐻_𝑗,𝑖) <max(𝐻_𝑗,𝑖) + 𝛾_𝑗𝜆(𝐻_𝑗,𝑖) < 𝑏, for0 ≤ 𝑖 ≤ 𝑛 − 1,

(v) max_{1≤𝑗≤𝑘}{𝛾_𝑗, (𝑏 − 𝑎)𝛾_𝑗} < (1/8)[min_{0≤𝑖≤𝑛, 1≤𝑗≤𝑘}{𝜖/𝑘,

𝜆(𝑓𝑖(𝐻𝑗)), 𝛽𝑗,𝑖}], where 𝛽𝑗,𝑖 = inf{|𝑓󸀠(𝑥)| : 𝑥 ∈

𝑓𝑖(𝐻_𝑗)}.

Let𝑥₀ = 𝑡_𝑖for some𝑖,𝛾, and𝐽 = [𝑐, 𝑑]be the associated

𝛾_𝑖and𝐻_𝑖, respectively. Define

𝑔_𝐽,𝑥₀_,𝛾(𝑥) = { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { −𝛾

3, if𝑥 = 𝑐 + 𝑥₀− 𝑐

2 ,

0, if𝑥 = 𝑐 +3 (𝑥0− 𝑐)

4 , 𝛾, if𝑥 = 𝑥₀,

0, if𝑥 = 𝑥₀+𝑑 − 𝑥0

4 ,

−𝛾₃, if𝑥 = 𝑥₀+𝑑 − 𝑥0

2 , 0, if𝑥 ∈ [𝑎, 𝑏] \ (𝑐, 𝑑) ,

linear, otherwise,

𝑟_𝐽,𝑥₀_,𝛾(𝑥) = { { { { { { { { { { { { { { { { { { { { { { {

𝛾, if 𝑥 = 𝑐 +𝑥0− 𝑐

4 , 0, if 𝑥 = 𝑥₀,

−𝛾 (𝑥₀− 𝑐)

𝑑 − 𝑥₀ if 𝑥 = 𝑥0+3 (𝑑 − 𝑥₄ 0),

0 if 𝑥 ∈ [𝑎, 𝑏] \ (𝑐, 𝑑) ,

linear, otherwise,

ℎ1(𝑥) = ∫ 𝑥

𝑎 𝑔𝐽,𝑥0,𝛾(𝑡) 𝑑𝑡,

ℎ2(𝑥) = ∫ 𝑥

𝑎 𝑟𝐽,𝑥0,𝛾(𝑡) 𝑑𝑡.

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We haveℎ_𝑚(𝑐) = ℎ_𝑚(𝑑) = ℎ󸀠_𝑚(𝑐) = ℎ󸀠_𝑚(𝑑) = 0for𝑚 = 1, 2,

ℎ₁(𝑥₀) = 0,ℎ󸀠₁(𝑥₀) = 𝛾,ℎ₂(𝑥₀) = (𝑥₀− 𝑐)𝛾/2, andℎ󸀠₂(𝑥₀) = 0. By considering four different cases, we construct a function

ℎ ∈ 𝐶1(𝐼, 𝐼) ∩ 𝐵(𝑓, 𝜖)so that, for Cases A and B,𝐹_𝑛(ℎ) ∩ {𝑥 :

(ℎ𝑛₎󸀠_{(𝑥) = 1} ∩ 𝐽 = 0}_and_𝐹

𝑛(ℎ) ∩ {𝑥 : (ℎ𝑛)󸀠(𝑥) = 1} ∩ 𝐽 =

{𝑥₀}for Cases C and D. From conditions (iv) and (v) we have

ℎ(𝐼) ⊂ 𝐼, and from condition (v) it follows thatℎis of the

same monotone type as𝑓on each𝑓𝑠(𝐽), 0 ≤ 𝑠 < 𝑛.

Case A. Let𝑓𝑛(𝑥) < 𝑥for𝐽 \ {𝑥₀}and𝑓𝑛(𝑥₀) = 𝑥₀.

(i) If 𝑓 is strictly increasing on 𝐽, then by taking

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andℎ(𝑥) = 𝑓(𝑥)on𝐼 \ 𝐽𝑜 and the function ℎ𝑛−1 is

also strictly increasing on𝑓(𝐽), and as a resultℎ𝑛(𝑥) <

𝑓𝑛_{(𝑥) ≤ 𝑥}_on_𝐽𝑜_{. Thus}_𝐹

𝑛(ℎ) ∩ 𝐽 = 0.

(ii) If𝑓is strictly decreasing on𝐽, then by takingℎ(𝑥) =

𝑓(𝑥) + ℎ₂(𝑥)we haveℎ(𝑥) > 𝑓(𝑥)on𝐽𝑜andℎ(𝑥) =

𝑓(𝑥)on𝐼 \ 𝐽𝑜 and the functionℎ𝑛−1 is also strictly

decreasing on𝑓(𝐽)and as a result,ℎ𝑛(𝑥) < 𝑓𝑛(𝑥) ≤ 𝑥

on𝐽𝑜. Thus𝐹_𝑛(ℎ) ∩ 𝐽 = 0.

Case B. Let𝑓𝑛(𝑥) > 𝑥for𝐽 \ {𝑥₀}and𝑓𝑛(𝑥₀) = 𝑥₀.

If𝑓is strictly increasing on𝐽, takeℎ(𝑥) = 𝑓(𝑥) + ℎ₂(𝑥),

and if𝑓is strictly decreasing on𝐽, takeℎ(𝑥) = 𝑓(𝑥) − ℎ₂(𝑥).

Then similar to Case A, we may show thatℎ𝑛(𝑥) > 𝑓𝑛(𝑥) ≥ 𝑥

for𝑥 ∈ 𝐽𝑜; hence,𝐹_𝑛(ℎ) ∩ 𝐽 = 0.

Case C. Let𝑓𝑛(𝑥) < 𝑥for𝑐 < 𝑥 < 𝑥₀,𝑓𝑛(𝑥) > 𝑥for𝑥₀< 𝑥 <

𝑑, and𝑓𝑛(𝑥₀) = 𝑥₀.

(i) If𝑓is strictly increasing on𝐽, then by takingℎ(𝑥) =

𝑓(𝑥) + ℎ₁(𝑥)we haveℎ(𝑥) < 𝑓(𝑥)for𝑐 < 𝑥 < 𝑥₀ andℎ(𝑥) > 𝑓(𝑥)for𝑥₀ < 𝑥 < 𝑑, and the function

ℎ𝑛−1_{is also strictly increasing on}_𝑓(𝐽)_{. Thus}_ℎ𝑛_{(𝑥) <}

𝑓𝑛_{(𝑥) < 𝑥}_for_{𝑐 < 𝑥 < 𝑥}

0andℎ𝑛(𝑥) > 𝑓𝑛(𝑥) > 𝑥for

𝑥₀< 𝑥 < 𝑑; hence,𝐹_𝑛(ℎ) ∩ 𝐽 = {𝑥₀}.

(ii) If𝑓is strictly decreasing on𝐽, then by takingℎ(𝑥) =

𝑓(𝑥) − ℎ₁(𝑥)we haveℎ(𝑥) > 𝑓(𝑥)for𝑐 < 𝑥 < 𝑥₀ andℎ(𝑥) < 𝑓(𝑥)for𝑥₀ < 𝑥 < 𝑑, and the function

ℎ𝑛−1_{is also decreasing on}_𝑓(𝐽)_{. Thus}_ℎ𝑛_{(𝑥) < 𝑓}𝑛_{(𝑥) <}

𝑥for𝑐 < 𝑥 < 𝑥₀andℎ𝑛(𝑥) > 𝑓𝑛(𝑥) > 𝑥for𝑥₀< 𝑥 <

𝑑; hence,𝐹_𝑛(ℎ) ∩ 𝐽 = {𝑥₀}.

Case D. Let𝑓𝑛(𝑥) > 𝑥for𝑐 < 𝑥 < 𝑥₀,𝑓𝑛(𝑥) < 𝑥for𝑥₀< 𝑥 <

𝑑, and𝑓𝑛(𝑥₀) = 𝑥₀.

If𝑓is strictly increasing on𝐽, takeℎ(𝑥) = 𝑓(𝑥) − ℎ₁(𝑥),

and if𝑓is strictly decreasing on𝐽, takeℎ(𝑥) = 𝑓(𝑥) + ℎ₁(𝑥).

Then similar to Case C, we may show thatℎ𝑛(𝑥) > 𝑓𝑛(𝑥) > 𝑥

for𝑐 < 𝑥 < 𝑥₀andℎ𝑛(𝑥) < 𝑓𝑛(𝑥) < 𝑥for𝑥₀ < 𝑥 < 𝑑. Thus

𝐹_𝑛(ℎ) ∩ 𝐽 = {𝑥₀}.

Note thatℎ𝑛is strictly increasing on𝐽 ∪ 𝑓𝑛(𝐽), so we have

𝐹_𝑛(ℎ) ∩ [(𝐽 \ 𝑓𝑛(𝐽)) ∪ (𝑓𝑛(𝐽) \ 𝐽)] = 0. (14)

Thus for𝑥 ∈ 𝐼either orb(𝑥, ℎ) = orb(𝑥, 𝑓)or orb(𝑥, ℎ) ∩

𝐽 ̸= 0, of which in such case𝐹_𝑛(ℎ)∩𝐽 = {𝑥₀},ℎ󸀠(𝑥₀) = 𝑓󸀠(𝑥₀)±

𝛾, andℎ󸀠(ℎ𝑖(𝑥₀)) = 𝑓󸀠(𝑓𝑖(𝑥₀))for1 ≤ 𝑖 ≤ 𝑛 − 1. Thus

(ℎ𝑛)󸀠(𝑥₀)

= ℎ󸀠(ℎ𝑛−1(𝑥0)) ℎ󸀠(ℎ𝑛−2(𝑥0)) ⋅ ⋅ ⋅ ℎ󸀠(ℎ (𝑥0)) ℎ󸀠(𝑥0)

= 𝑓󸀠(𝑓𝑛−1(𝑥₀)) 𝑓󸀠(𝑓𝑛−2(𝑥₀))

× 𝑓󸀠(𝑓𝑛−3(𝑥₀)) ⋅ ⋅ ⋅ 𝑓󸀠(𝑓 (𝑥₀)) ℎ󸀠(𝑥₀)

= (𝑓𝑛)

󸀠_(𝑥

0) ℎ󸀠(𝑥0)

𝑓󸀠_(𝑥

0) =

𝑓󸀠_(𝑥 0) ± 𝛾

𝑓󸀠_(𝑥 0)

= [1 ± 𝛾 𝑓󸀠_(𝑥

0)] ̸= 1.

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We have

𝐹_𝑛(ℎ) ∩ {𝑥 : (ℎ𝑛)󸀠(𝑥) = 1}

= [𝐹_𝑛(𝑓) ∩ {𝑥 : (𝑓𝑛)󸀠(𝑥) = 1}] \ {𝑥₀} . (16)

Also

𝜌₀(𝑓󸀠, ℎ󸀠) ≤sup

𝑥∈𝐽{󵄨󵄨󵄨󵄨𝑔 (𝑥)󵄨󵄨󵄨󵄨,|𝑟(𝑥)|} ≤ 𝜖8𝑘,

𝜌₀(𝑓, ℎ) ≤sup

𝑥∈𝐽{󵄨󵄨󵄨󵄨ℎ1(𝑥)󵄨󵄨󵄨󵄨 ,󵄨󵄨󵄨󵄨ℎ2(𝑥)󵄨󵄨󵄨󵄨} ≤

𝜖 8𝑘.

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Hence𝜌₁(𝑓, ℎ) ≤ 𝜌₀(𝑓, ℎ) + 𝜌₀(𝑓󸀠, ℎ󸀠) ≤ 𝜖/4𝑘.

Doing this recursively for each𝑡_𝑗,1 ≤ 𝑗 ≤ 𝑘, we get a

function𝑔 ∈ 𝐵(𝑓, 𝜖/4)such that𝐹_𝑛(𝑔) ∩ {𝑥 : (𝑔𝑛)󸀠(𝑥) = 1} =

0.

Theorem 8. Typically continuously differentiable self-maps of intervals have a countable set of periodic points.

Proof. Take

𝐻𝑛= {𝑓 ∈ 𝐶1(𝐼, 𝐼) : 𝐹𝑛(𝑓) is not finite} ,

𝐵1= {𝑓 ∈ 𝐶1(𝐼, 𝐼) : 𝑎 ∈ 𝑃𝑓2∩ {𝑥 : (𝑓2) 󸀠

(𝑥) = 1}

with𝑓 (𝑎) = 𝑏} ,

𝐹₁= {𝑓 ∈ 𝐶1(𝐼, 𝐼) : ∃𝑥₀∈ 𝐼 such that

𝑥0∈ 𝐹1(𝑓) and𝑓󸀠(𝑥0) = 1} ,

E_𝑛= {𝑓 ∈ 𝐶1_{(𝐼, 𝐼) : 𝐹}

𝑛(𝑓) ∩ {𝑥 : (𝑓𝑛)󸀠(𝑥) = 1} ̸= 0} .

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The sets𝐹₁and𝐵₁are first category sets (see Theorems

5and6). For𝑛 ≥ 1from Lemmas4and1we have thatE_𝑛

is closed and𝐻_𝑛 ⊆ E_𝑛. Without loss of generality we may

assume that𝑓 ∈ 𝐶1(𝐼, 𝐼)is neither a constant function nor

a polynomial of first degree, and then by Lemma2we may

choose a polynomial 𝑔 ∈ 𝐶1(𝐼, 𝐼) of degree𝑛 ≥ 2 such

that𝑔(𝐼) ⊂ 𝐼𝑜and𝜌₁(𝑓, 𝑔) < 𝜖/4. Using Theorem7we can

construct ℎ ∈ 𝐶1(𝐼, 𝐼)so that 𝜌₁(𝑔, ℎ) < 𝜖/4and the set

𝐴𝑛 = 𝐹𝑛(ℎ) ∩ {𝑥 : (ℎ𝑛)󸀠(𝑥) = 1} = 0, thus𝜌1(𝑓, ℎ) < 𝜖

andℎ ∉ E_𝑛, henceE𝑜_𝑛 = 0. This implies that for each𝑘 ≥ 2

the setE_𝑘is nowhere dense. Thus𝐹 = (∪∞_𝑘=2E_𝑘) ∪ 𝐹₁∪ 𝐵₁is

a first category set, so𝑀 = 𝐶1(𝐼, 𝐼) \ 𝐹is a residual set, and,

for𝑓 ∈ 𝑀,𝑃(𝑓) = ∪∞_𝑛=1𝐹_𝑛(𝑓)is countable.

Acknowledgments

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[1] S. J. Agronsky, A. M. Bruckner, and M. Laczkovich, “Dynamics

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[2] A. A. Alikhani-Koopaei, “On the size of the sets of chain recurrent points, periodic points, and fixed points of functions,”

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[3] A. A. Alikhani-Koopaei, “On periodic and recurrent points

of continuous functions,” inProceedings of the 28th Summer

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[4] A. Alikhani-Koopaei, “On common fixed points, periodic

points, and recurrent points of continuous functions,”

Interna-tional Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 39, pp. 2465–2473, 2003.

[5] T. H. Steele, “A note on periodic points and commuting

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[6] A. Alikhani-Koopaei, “On common fixed and periodic points

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