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; axa12@psu.edu
Received 6 December 2012; Accepted 16 February 2013
Academic Editor: Paolo Ricci
Copyright Β© 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 βbigβ or βsmallβ 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 [1β5]). 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. Schwartzβs
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
πΉ1(π) = {π₯ : π(π₯) = π₯}denotes the set of fixed points ofπ.
Forπandπ βNwe defineππ(π₯)by induction:
ππ(π₯) = π (ππβ1(π₯))
withπ0(π₯) = π₯, π1(π₯) = π (π₯) .
(1)
The orbit ofπ₯underπis given by the sequence orb(π₯, π) =
{ππ(π₯)}β
π=0. Forπ βN, letπΉπ(π) = {π₯ : ππ(π₯) = π₯}andπππbe
the set of periodic points of orderπ; that is,
πππ= πΉπ(π) \ βͺπ<ππΉπ(π) . (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 πof the
interval[π, π],βπβis the length of the largest subinterval of
π,π΅(π, π)is the open ball aboutπwith radiusπ,π΄βdenotes
the set of interior points ofπ΄, andπ(πΌ)is the length of the
intervalπΌ.
2. Continuously Differentiable Functions
fromπΌinto itself, respectively. Recall that the usual metricsπ0 andπ1onπΆ(πΌ, πΌ)andπΆ1(πΌ, πΌ), respectively, are given by
π0(π, π) =sup
π₯βπΌσ΅¨σ΅¨σ΅¨σ΅¨π(π₯) β π(π₯)σ΅¨σ΅¨σ΅¨σ΅¨ forπ, π β πΆ (πΌ, πΌ) ,
π1(π, π) = π0(π, π) + π0(πσΈ , πσΈ ) forπ, π β πΆ1(πΌ, πΌ) . (3)
It is well known that the metric spaces (πΆ(πΌ, πΌ); π0)and
(πΆ1(πΌ, πΌ); π
1)are complete and hence Baireβs category theorem
holds in these spaces. We say that a typical function inπΆ(πΌ, πΌ)
orπΆ1(πΌ, πΌ)has a certain property if the set of those functions
which does not have this property is of first category inπΆ(πΌ, πΌ)
or inπΆ1(πΌ, πΌ). (Some authors prefer using the term generic
instead of typical.) It is known that typically continuous
self-maps of an interval have π-perfect, measure zero sets
of periodic points (see [2]). Here we show that typically
members of(πΆ1(πΌ, πΌ), π1)have a finite set of fixed points and
a countable set of periodic points.
Lemma 1. Letπ β₯ 1andπ β πΆ1(πΌ, πΌ)so thatπΉπ(π)is not
finite. Then there exists π₯0 β πΌ so thatππ(π₯0) = π₯0 and
(ππ)σΈ (π₯ 0) = 1.
Proof. SupposeπΉπ(π)is not finite, and then we can choose
a strictly monotone sequence in πΉπ(π). Without loss of
generality, assume {π₯π}βπ=1 β πΉπ(π)so that π₯π < π₯π+1 for
each π β₯ 1. Let β(π₯) = ππ(π₯) β π₯, and then, for each π,
β(π₯π) = β(π₯π+1) = 0. Thus by Rolleβs Theorem we haveπ¦π,
π₯π < π¦π < π₯π+1so thatβσΈ (π¦π) = 0. Letπ₯0 = limπ β βπ₯π, then
π₯0 β πΌ, and limπ β βπ¦π = π₯0, and from the continuity ofππ
and(ππ)σΈ it follows thatππ(π₯0) = π₯0and(ππ)σΈ (π₯0) = 1.
Lemma 2. For eachπ β πΆ1(πΌ, πΌ)and0 < π < 1there is a
polynomialπ : πΌ β πΌπwithπ1(π, π) < π.
Proof. Letπ = supπ₯βπΌ{|πσΈ (π₯)|, |π|, |π|, 1}whereπΌ = [π, π].
Takeπ½ = π/8π,πΌ = π½(π + π)/2, andπ(π₯) = πΌ + (1 β π½)π(π₯).
It is easy to see that0 < π½ < 1,π(πΌ) β πΌπ, andπ1(π, π) < π/4.
Let
π1= 12π₯β[π,π]inf {π (π₯) β π, π β π (π₯) ,8 (π β π)π ,8π} (4)
andπ (π‘)be a polynomial withπ0(πσΈ , π ) < π1/(π β π + 1). Then
π(π₯) = π(π) + β«ππ₯π (π‘)ππ‘is a polynomial andπ(π₯) β π(π₯) =
β«ππ₯(πσΈ (π‘) β π (π‘))ππ‘. Thus we have
π1(π, π) = π0(πσΈ , π ) + π0(π, π) β€ π1
(π β π) + 1+
π1(π β π) (π β π) + 1
= π1< π4,
(5)
henceπ1(π, π) β€ π1(π, π) + π1(π, π) < π/2. Fromπ(πΌ) β [π +
2π1, π β 2π1]andπ1(π, π) β€ π1it follows thatπ(πΌ) β πΌπ.
Lemma 3. Letπ > 2be a positive integer andπ β πΆ1(πΌ, πΌ).
Ifπ₯0is a periodic point of order πwith(ππ)σΈ (π₯0) = 1, then
orb(π₯0, π) β© {π, π} = 0.
Proof. We show thatπ βorb(π₯0, π). The caseπ β orb(π₯0, π)
follows similarly. Let orb(π₯0, π) = {π = π₯0, π₯1, . . . , π₯π = π₯0}
withπ₯π = π(π₯πβ1)for1 β€ π β€ π. Sinceπ > 2, there exist
π₯π β orb(π₯0, π) β© (π, π)and1 β€ π β€ π β 1so thatππ (π₯π) =
π. Sinceππ β πΆ1(πΌ, πΌ)and(ππ )σΈ (π₯0) ΜΈ= 0, there existsπΏ1 > 0
so thatπ < π₯πβ πΏ1 < π₯π < π₯π+ πΏ1 < πand ππ is strictly
increasing or strictly decreasing onπ½ = [π₯πβ πΏ1, π₯π+ πΏ1]. Let
ππ be strictly increasing onπ½, and then forπ₯
πβπΏ1β€ π₯ β€ π₯πwe
haveππ (π₯) β€ ππ (π₯π) = π; hence,ππ |[π₯πβπΏ1,π₯π]= π, implying that
(ππ )σΈ (π₯
π) = 0, a contradiction to(ππ)σΈ (π₯0) = 1. On the other
hand whenππ is strictly decreasing onπ½, forπ₯πβ€ π₯ β€ π₯π+ πΏ1
we haveππ (π₯) β€ ππ (π₯π) = π; hence,ππ |[π₯π,π₯π+πΏ1]= πimplying
that(ππ )σΈ (π₯π) = 0, a contradiction to(ππ)σΈ (π₯0) = 1.
Lemma 4. Forπ β₯ 1, the set
Eπ= {π β πΆ1(πΌ, πΌ) : πΉπ(π) β© {π₯ : (ππ)σΈ (π₯) = 1} ΜΈ= 0}
(6)
is closed inπΆ1(πΌ, πΌ).
Proof. Letππ β Eπ and π1(ππ, π) β 0. Then there exists
{π‘π}βπ=1 β πΌsuch thatπππ(π‘π) = π‘πand(πππ)σΈ (π‘π) = 1. Without
loss of generality, we may assume that limπ β βπ‘π = π‘0, then
π‘0β πΌ. Letπ > 0be arbitrary. Due to the uniform continuity of
ππonπΌthere exists a positive integerπ
1such that, forπ β₯ π1,
|ππ(π‘π) β ππ(π‘0)| < π. Since π0(ππ, π) β 0, there exists a
positive integerπ2 such that, for π β₯ π2,π0(πππ, ππ) < π.
Choose the positive integerπ3so that|π‘πβ π‘0| < πforπ β₯ π3,
and letπ1=max(π1, π2, π3). Then forπ β₯ π1we have
σ΅¨σ΅¨σ΅¨σ΅¨
σ΅¨ππ(π‘0) β π‘0σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨
β€ σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨ππ(π‘0) β ππ(π‘π)σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨ +σ΅¨πσ΅¨σ΅¨σ΅¨σ΅¨ π(π‘π) β πππ (π‘π)σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨πππ(π‘π) β π‘0σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨
β€ π + π0(ππ, πππ) + σ΅¨σ΅¨σ΅¨σ΅¨π‘πβ π‘0σ΅¨σ΅¨σ΅¨σ΅¨ β€ 3π,
(7)
Implying thatππ(π‘0) = π‘0. We also have
σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨(πππ)
σΈ
(π‘π) β (ππ)σΈ (π‘π)σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨σ΅¨ β€π β βlimπ1(πππ, ππ) = 0. (8)
Thus we have limπ β β|1 β (ππ)σΈ (π‘π)| = |1 β (ππ)σΈ (π‘0)| = 0and
π βEπ.
Theorem 5. There exists a residual subsetπ1ofπΆ1(πΌ, πΌ)such
that, for everyπ β π1,πΉ1(π)is finite.
Proof. If πΉ1(π) has infinitely many elements, then from
Lemma1it follows that there exists a pointπ₯0 β πΉ1(π)so
thatπσΈ (π₯0) = 1. Let
E1= {π β πΆ1(πΌ, πΌ) : βπ₯0β πΌ β© πΉ1(π) , πσΈ (π₯0) = 1} . (9)
From Lemma4the setE1is closed inπΆ1(πΌ, πΌ). To show that
Lemma2, there exists a polynomialπsuch thatπ1(π, π) < π/2 andπ(πΌ) β [π + π1, π β π1]for some0 < π1< π/2. Let
π (π ) = {π₯ : πσΈ (π₯) = 1} = {π‘1, π‘2, . . . , π‘π} . (10)
Choose0 < π2< π1so thatπ(π₯) β π₯ ΜΈ= π2for allπ₯ β π(π )and
takeβ = π β π2. Thenβ β πΆ1(πΌ, πΌ),π1(β, π) < π,π(β) = {π₯ :
βσΈ (π₯) = 1} = π(π), andβ(π₯) ΜΈ= π₯onπ(β), implying thatE 1
is of first category andπ1= πΆ1(πΌ, πΌ) \E1is residual.
Theorem 6. The set of functionsπ β πΆ1(πΌ, πΌ)with π(π) =
π, π(π) = π, and(π2)σΈ (π) = 1is a nowhere dense subset of
πΆ1(πΌ, πΌ).
Proof. Let π΄ = {π β πΆ1(πΌ, πΌ) : π(π) = π, π(π) =
π, and (π2)σΈ (π) = 1}. It is easy to see thatπ΄is a closed subset ofπΆ1(πΌ, πΌ). To show thatπ΄is nowhere dense, letπ β πΆ1(πΌ, πΌ) andπ > 0. Choose0 < πΎ < min{π/(4(π β π) + 4), 1}and
π < π < πsuch that|π(π‘) β π(π)| < (π β π)[1 β π/4] for
π‘ β [π, π]. Letβ(π₯) = π(π₯) + β«ππ₯π(π‘)ππ‘, where
π (π‘) = { { { { { { { { { { { { { { { { { { { { { { {
βπΎ, π‘ = π,
0, π‘ = π +π β π 4 , πΎ
3, π‘ = π + (π β π)2 , 0, π‘ = π, π‘ = π,
linear, otherwise.
(11)
It is clear thatβ(π) = π(π) = π,β(π) = π(π) = π,βσΈ (π) =
πσΈ (π) β πΎ, andβσΈ (π) = πσΈ (π). It is easy to see thatβ β πΆ1(πΌ, πΌ),
(β2)σΈ (π) = (β2)σΈ (π) ΜΈ= 1, andπ
1(π, β) < π/2.
Theorem 7. Letπ > 0,π β πΆ1(πΌ, πΌ), andπ΄π = {π₯ : (ππ)σΈ (π₯) =
1} β© πΉπ(π)be a finite set, andπππ(π₯, π) β (π, π)forπ₯ β π΄π.
Then there exists a functionπ β πΆ1(πΌ, πΌ)withπ1(π, π) < π/4
such that{π₯ : (ππ)σΈ (π₯) = 1} β© πΉπ(π) = 0.
Proof. Letπ΄π = {π₯ : (ππ)σΈ (π₯) = 1} β© πΉπ(π) = {π§π}ππ=1. Let
π΅ = {π‘π}ππ=1be the elements ofπ΄πwith distinct orbits, that is,
orb(π‘π, π) β©orb(π‘π, π) = 0forπ ΜΈ= πandβͺππ=1orb(π‘π, π) β π΄π.
It is clear that eachπ‘π β π΅is a periodic point ofπwith some
periodπwhereπis a factor ofπ. This suggests that if one
can construct a functionπthat is sufficiently close toπ, and
eitherπ‘πβ πππor(ππ)σΈ (π‘π) ΜΈ= 1, then
{π₯ : (ππ)σΈ (π₯) = 1} β© πΉπ(π) β π΄π\orb(π‘π, π) . (12)
By repeating this process for eachπ‘π β π΅in a finite number
of steps we can construct the desired functionπ. Thus for
convenience, we assume that, for1 β€ π β€ π,π§π β πππ.
Consider the partitionπof [π, π]obtained by {ππ(π‘π), 1 β€
π β€ π, 1 β€ π β€ π},π1 = (1/4)min{π, β π β}, and choose a
positiveπΏless thanπ1such that, for eachπ, 1 β€ π β€ πand each
π, 1 β€ π β€ π, and if|π‘βπ‘π| < πΏ, then|ππ(π‘)βππ(π‘π)| < π1. Given
that, forπ₯ β π΄π, orb(π₯, π) β (π, π),(ππ)σΈ is continuous, and
(ππ)σΈ (π₯) = 1onπ΅, we may choose the nondegenerate closed
intervalsπ»π, 1 β€ π β€ πof length less thanπΏand the positive
numbersπΎπsuch that
(i)π‘π β (π»π)β is the midpoint of π»π and ππ(π»π) β©
orb(π‘π, π) = {ππ(π‘π)}for0 β€ π β€ π,
(ii)πis strictly monotone on eachπ»π,π = ππ(π»π)for0 β€
π β€ π β 1,1 β€ π β€ π, whereπ»π,0= π»π,
(iii) the intervalsπ»π,π = ππ(π»π)are mutually disjoint for
1 β€ π β€ π,1 β€ π β€ π,
(iv)π <min(π»π,π) β πΎππ(π»π,π) <max(π»π,π) + πΎππ(π»π,π) < π, for0 β€ π β€ π β 1,
(v) max1β€πβ€π{πΎπ, (π β π)πΎπ} < (1/8)[min0β€πβ€π, 1β€πβ€π{π/π,
π(ππ(π»π)), π½π,π}], where π½π,π = inf{|πσΈ (π₯)| : π₯ β
ππ(π»π)}.
Letπ₯0 = π‘πfor someπ,πΎ, andπ½ = [π, π]be the associated
πΎπandπ»π, respectively. Define
ππ½,π₯0,πΎ(π₯) = { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { βπΎ
3, ifπ₯ = π + π₯0β π
2 ,
0, ifπ₯ = π +3 (π₯0β π)
4 , πΎ, ifπ₯ = π₯0,
0, ifπ₯ = π₯0+π β π₯0
4 ,
βπΎ3, ifπ₯ = π₯0+π β π₯0
2 , 0, ifπ₯ β [π, π] \ (π, π) ,
linear, otherwise,
ππ½,π₯0,πΎ(π₯) = { { { { { { { { { { { { { { { { { { { { { { {
πΎ, if π₯ = π +π₯0β π
4 , 0, if π₯ = π₯0,
βπΎ (π₯0β π)
π β π₯0 if π₯ = π₯0+3 (π β π₯4 0),
0 if π₯ β [π, π] \ (π, π) ,
linear, otherwise,
β1(π₯) = β« π₯
π ππ½,π₯0,πΎ(π‘) ππ‘,
β2(π₯) = β« π₯
π ππ½,π₯0,πΎ(π‘) ππ‘.
(13)
We haveβπ(π) = βπ(π) = βσΈ π(π) = βσΈ π(π) = 0forπ = 1, 2,
β1(π₯0) = 0,βσΈ 1(π₯0) = πΎ,β2(π₯0) = (π₯0β π)πΎ/2, andβσΈ 2(π₯0) = 0. By considering four different cases, we construct a function
β β πΆ1(πΌ, πΌ) β© π΅(π, π)so that, for Cases A and B,πΉπ(β) β© {π₯ :
(βπ)σΈ (π₯) = 1} β© π½ = 0andπΉ
π(β) β© {π₯ : (βπ)σΈ (π₯) = 1} β© π½ =
{π₯0}for Cases C and D. From conditions (iv) and (v) we have
β(πΌ) β πΌ, and from condition (v) it follows thatβis of the
same monotone type asπon eachππ (π½), 0 β€ π < π.
Case A. Letππ(π₯) < π₯forπ½ \ {π₯0}andππ(π₯0) = π₯0.
(i) If π is strictly increasing on π½, then by taking
andβ(π₯) = π(π₯)onπΌ \ π½π and the function βπβ1 is
also strictly increasing onπ(π½), and as a resultβπ(π₯) <
ππ(π₯) β€ π₯onπ½π. ThusπΉ
π(β) β© π½ = 0.
(ii) Ifπis strictly decreasing onπ½, then by takingβ(π₯) =
π(π₯) + β2(π₯)we haveβ(π₯) > π(π₯)onπ½πandβ(π₯) =
π(π₯)onπΌ \ π½π and the functionβπβ1 is also strictly
decreasing onπ(π½)and as a result,βπ(π₯) < ππ(π₯) β€ π₯
onπ½π. ThusπΉπ(β) β© π½ = 0.
Case B. Letππ(π₯) > π₯forπ½ \ {π₯0}andππ(π₯0) = π₯0.
Ifπis strictly increasing onπ½, takeβ(π₯) = π(π₯) + β2(π₯),
and ifπis strictly decreasing onπ½, takeβ(π₯) = π(π₯) β β2(π₯).
Then similar to Case A, we may show thatβπ(π₯) > ππ(π₯) β₯ π₯
forπ₯ β π½π; hence,πΉπ(β) β© π½ = 0.
Case C. Letππ(π₯) < π₯forπ < π₯ < π₯0,ππ(π₯) > π₯forπ₯0< π₯ <
π, andππ(π₯0) = π₯0.
(i) Ifπis strictly increasing onπ½, then by takingβ(π₯) =
π(π₯) + β1(π₯)we haveβ(π₯) < π(π₯)forπ < π₯ < π₯0 andβ(π₯) > π(π₯)forπ₯0 < π₯ < π, and the function
βπβ1is also strictly increasing onπ(π½). Thusβπ(π₯) <
ππ(π₯) < π₯forπ < π₯ < π₯
0andβπ(π₯) > ππ(π₯) > π₯for
π₯0< π₯ < π; hence,πΉπ(β) β© π½ = {π₯0}.
(ii) Ifπis strictly decreasing onπ½, then by takingβ(π₯) =
π(π₯) β β1(π₯)we haveβ(π₯) > π(π₯)forπ < π₯ < π₯0 andβ(π₯) < π(π₯)forπ₯0 < π₯ < π, and the function
βπβ1is also decreasing onπ(π½). Thusβπ(π₯) < ππ(π₯) <
π₯forπ < π₯ < π₯0andβπ(π₯) > ππ(π₯) > π₯forπ₯0< π₯ <
π; hence,πΉπ(β) β© π½ = {π₯0}.
Case D. Letππ(π₯) > π₯forπ < π₯ < π₯0,ππ(π₯) < π₯forπ₯0< π₯ <
π, andππ(π₯0) = π₯0.
Ifπis strictly increasing onπ½, takeβ(π₯) = π(π₯) β β1(π₯),
and ifπis strictly decreasing onπ½, takeβ(π₯) = π(π₯) + β1(π₯).
Then similar to Case C, we may show thatβπ(π₯) > ππ(π₯) > π₯
forπ < π₯ < π₯0andβπ(π₯) < ππ(π₯) < π₯forπ₯0 < π₯ < π. Thus
πΉπ(β) β© π½ = {π₯0}.
Note thatβπis strictly increasing onπ½ βͺ ππ(π½), so we have
πΉπ(β) β© [(π½ \ ππ(π½)) βͺ (ππ(π½) \ π½)] = 0. (14)
Thus forπ₯ β πΌeither orb(π₯, β) = orb(π₯, π)or orb(π₯, β) β©
π½ ΜΈ= 0, of which in such caseπΉπ(β)β©π½ = {π₯0},βσΈ (π₯0) = πσΈ (π₯0)Β±
πΎ, andβσΈ (βπ(π₯0)) = πσΈ (ππ(π₯0))for1 β€ π β€ π β 1. Thus
(βπ)σΈ (π₯0)
= βσΈ (βπβ1(π₯0)) βσΈ (βπβ2(π₯0)) β β β βσΈ (β (π₯0)) βσΈ (π₯0)
= πσΈ (ππβ1(π₯0)) πσΈ (ππβ2(π₯0))
Γ πσΈ (ππβ3(π₯0)) β β β πσΈ (π (π₯0)) βσΈ (π₯0)
= (ππ)
σΈ (π₯
0) βσΈ (π₯0)
πσΈ (π₯
0) =
πσΈ (π₯ 0) Β± πΎ
πσΈ (π₯ 0)
= [1 Β± πΎ πσΈ (π₯
0)] ΜΈ= 1.
(15)
We have
πΉπ(β) β© {π₯ : (βπ)σΈ (π₯) = 1}
= [πΉπ(π) β© {π₯ : (ππ)σΈ (π₯) = 1}] \ {π₯0} . (16)
Also
π0(πσΈ , βσΈ ) β€sup
π₯βπ½{σ΅¨σ΅¨σ΅¨σ΅¨π (π₯)σ΅¨σ΅¨σ΅¨σ΅¨,|π(π₯)|} β€ π8π,
π0(π, β) β€sup
π₯βπ½{σ΅¨σ΅¨σ΅¨σ΅¨β1(π₯)σ΅¨σ΅¨σ΅¨σ΅¨ ,σ΅¨σ΅¨σ΅¨σ΅¨β2(π₯)σ΅¨σ΅¨σ΅¨σ΅¨} β€
π 8π.
(17)
Henceπ1(π, β) β€ π0(π, β) + π0(πσΈ , βσΈ ) β€ π/4π.
Doing this recursively for eachπ‘π,1 β€ π β€ π, we get a
functionπ β π΅(π, π/4)such thatπΉπ(π) β© {π₯ : (ππ)σΈ (π₯) = 1} =
0.
Theorem 8. Typically continuously differentiable self-maps of intervals have a countable set of periodic points.
Proof. Take
π»π= {π β πΆ1(πΌ, πΌ) : πΉπ(π) is not finite} ,
π΅1= {π β πΆ1(πΌ, πΌ) : π β ππ2β© {π₯ : (π2) σΈ
(π₯) = 1}
withπ (π) = π} ,
πΉ1= {π β πΆ1(πΌ, πΌ) : βπ₯0β πΌ such that
π₯0β πΉ1(π) andπσΈ (π₯0) = 1} ,
Eπ= {π β πΆ1(πΌ, πΌ) : πΉ
π(π) β© {π₯ : (ππ)σΈ (π₯) = 1} ΜΈ= 0} .
(18)
The setsπΉ1andπ΅1are first category sets (see Theorems
5and6). Forπ β₯ 1from Lemmas4and1we have thatEπ
is closed andπ»π β Eπ. Without loss of generality we may
assume thatπ β πΆ1(πΌ, πΌ)is neither a constant function nor
a polynomial of first degree, and then by Lemma2we may
choose a polynomial π β πΆ1(πΌ, πΌ) of degreeπ β₯ 2 such
thatπ(πΌ) β πΌπandπ1(π, π) < π/4. Using Theorem7we can
construct β β πΆ1(πΌ, πΌ)so that π1(π, β) < π/4and the set
π΄π = πΉπ(β) β© {π₯ : (βπ)σΈ (π₯) = 1} = 0, thusπ1(π, β) < π
andβ β Eπ, henceEππ = 0. This implies that for eachπ β₯ 2
the setEπis nowhere dense. ThusπΉ = (βͺβπ=2Eπ) βͺ πΉ1βͺ π΅1is
a first category set, soπ = πΆ1(πΌ, πΌ) \ πΉis a residual set, and,
forπ β π,π(π) = βͺβπ=1πΉπ(π)is countable.
Acknowledgments
References
[1] S. J. Agronsky, A. M. Bruckner, and M. Laczkovich, βDynamics
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[3] A. A. Alikhani-Koopaei, βOn periodic and recurrent points
of continuous functions,β inProceedings of the 28th Summer
Symposium Conference, Real Analysis Exchange, pp. 37β40, 2004.
[4] A. Alikhani-Koopaei, βOn common fixed points, periodic
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