Volume 2008, Article ID 194875,8pages doi:10.1155/2008/194875
Research Article
Periodic Orbits with Least Period Three
on the Circle
Xuezhi Zhao
Department of Mathematics, Capital Normal University, Beijing 100037, China
Correspondence should be addressed to Xuezhi Zhao,[email protected]
Received 19 August 2007; Accepted 3 December 2007
Recommended by J. R. L. Webb
The self-maps on the circle having periodic orbits with least period 3 are classified into relative homotopy-conjugacy classes. We will show that except for the maps in one exceptional class, all these kinds of maps have periodic orbits with all least periods but 2. Thus, a kind of Sharkovskii’s type theorem on the circle is obtained.
Copyrightq2008 Xuezhi Zhao. 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.
1. Introduction
Because of1, it is well known that a map on the interval must have periodic points of each
least period if it has a periodic point of least period 3. This result was actually a special case of Sharkovskii’s theorem2. Its various generalizations became a rich and active field in
mathe-maticssee3.
It is very natural to ask for an analogy of Sharkovskii’s theorem for maps on the circle. Unfortunately, the answer is negative if the period of a periodic orbit is given merely. For any positive integer n, the 360/n degree rotation admits all points as periodic points with least period n, but has no periodic point with any other least periods. Thus, one needs more information about the given periodic orbits in order to obtain other periodic points. In 4,
the existence of periodic points was obtained according to the homotopy classes, that is, the degrees, of maps on the circle. More general, combinatorial structures of given period orbits of maps on graphsas opposed to merely their periodswere formulated as the “pattern”5,6.
on the circle having periodic orbits with least period 3, we will show that except for the maps in one relative homotopy-conjugacy class, all maps have periodic points with least periodnfor all positive integern /2, where the given periodic orbits are regarded as their finite invariant subsets. Our main tool is the relative Nielsen fixed point theory. At the end, we present some examples showing that our conclusion is not true if the least period of the given periodic orbit is larger than 5.
This paper is organized as follows. InSection 2, we will review some results in relative Nielsen fixed point theory, which will be used here. A new classification of relative maps, namely, “relative homotopy-conjugacy classes” will be defined in Section 3. Our main result will be given inSection 4. InSection 5, we illustrate some examples when the given period is larger than 5.
2. Surplus fixed point classes
In this section, we will review some definitions and results in relative Nielsen fixed point the-ory, especially those related to our purpose here; see8for more details and9for general relative Nielsen fixed point theory.
Consider a mapf : X→Xon a compact connected polyhedronX. Letp : X→X be the universal covering ofX. A mapf: X→X is said to be alifting off if the following diagram commutes:
X
p
f
X
p
X f X
2.1
LetUbe a path-connected subset ofX. Fix a componentU ofp−1U, we have
U∩Fixf
f
p U∩Fixf, 2.2
where franges over all liftings of f. According to 8, Definition 2.1, two pairsf,U and f,U of f on U are said to be conjugate if there exists a covering translation γ such that
γU Uandfγ◦f◦γ−1.
Proposition 2.1. For any two pairsf,Uand f,Uoff on U, either pU ∩Fixf∩pU ∩
Fixf ∅, iff,Uandf,Uare not conjugate, orpU ∩Fixf pU∩Fixf, iff,U
andf,Uare conjugate.
The subset pU ∩Fixf of the fixed point set of f on U is said to be the fixed point class of fonUbeing determined by the pairf,U. By the above proposition, the fixed point set U∩Fixfof f onUsplits into a disjoint union of the fixed point classes off onU. Let
Definition 2.2see10, Definition 3.1and8, Definition 4.7. LetFbe a fixed point class off onX−Awhich is determined byf, C, whereCis a component ofp−1X−A. The fixed point classFis said to benonsurplusif there is a componentAofp−1AwithA∩clC/∅such that
fA⊂Ais said to besurplusif it is not nonsurplus.
Of most importance is that any surplus fixed point class is a compact set, and hence has a well-defined index. The so-called surplus Nielsen number SNf;X −Aof a relative map
f :X, A→X, AonX−Ais defined as the number of essential surplus fixed point classes of
fonX−A.
3. Relative homotopy-conjugacy classes
Here, we define a new classification of relative self maps, which is just a combination of relative homotopy classes and relative topological conjugacy classes.
Definition 3.1. Two relative maps f : X, A→X, A and f : X, A→X, A are said to be relatively homotopy-conjugate if there is a finite sequence of relative maps: fk : X, Ak
→X, Ak,k 0,1,2, . . . , n, withA0 A, An A,f0 f, andfn f such that for each k,
k1,2, . . . , n.
Either1fk−1andfk are relatively homotopic, that is,Ak−1 Akand there is a relative homotopyH :X×I, Ak×I→X, Aksuch thatHx,0 fk−1xandHx,1 fkxfor all
x∈X.
Or2fk−1andfkare relatively conjugate, that is, there is a relative homeomorphismhk such that the following diagram commutes:
X,Ak−1
hk
fk−1 X,Ak−1
hk
X,Ak fk X,Ak
3.1
Clearly, “relative homotopy-conjugacy” is an equivalent relation in the set of rela-tive maps on homeomorphic space pairs. Each equivalent class is said to be a relative homo topy conjugacy class. In the case of circle maps, a similar relation, a little restricted, was named as “h-equivalent”see5, page179.
Lemma 3.2. If two relative maps f : X, A→X, A and f : X, A→X, A are relatively
homotopy-conjugate. ThenSNf;X−A SNf;X−A.
Proof. It is sufficient to show that the surplus Nielsen number is invariant under relative
ho-motopy and relative conjugacy. The first part was proved in10, Theorem 3.6, and the second
part was given in11, Proposition 3.11.
Since any essential surplus fixed point class is always nonempty, we obtain the following theorem.
Theorem 3.3. Any relative map f : X, A→X, A which is relatively homotopyconjugate to a
This lower-bound property enables us to prove simultaneously the existence of fixed points for all maps in a given relative homotopy-conjugacy class, instead of proving their exis-tence individually.
4. Circle maps with periodic points of least period 3
In this section, we will consider the maps on the circle which have periodic points with least period 3. Let us fix some notations.
For a triplen0, n1, n2of integers, we define a relative mapfn0,n1,n2:S
1→S1by
fn0,n1,n2
eθiexp2πλ
n0,n1,n2
θ
2π
i
, 0≤θ <2π, 4.1
where the mapλn0,n1,n2: 0,1→R
1is defined by
λn0,n1,n2x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
6x1
3 if 0≤x≤
1 9,
9n0x−n01 if 1 9 ≤x≤
2 9, −3xn01 2
3 if
2 9 ≤x≤
1 3,
3xn0−1
3 if
1 3 ≤x≤
4 9,
9n1xn01−4n1 if 4 9 ≤x≤
5 9,
n0n11 if 5
9 ≤x≤ 7 9,
9n2xn0n11−7n2 if 7 9 ≤x≤
8 9,
3xn0n1n2−5
3 if
8
9 ≤x <1.
4.2
We may regardS1as a graph with three verticesυ
0, υ1, υ2and three edgesw0, w1, w2, where
wk {eθi |2kπ/3≤θ≤2k2π/3}andυk e2kπ/3i,k 0,1,2. Thus,fn0,n1,n2is a relative
self-map on the pairS1, V {υ0, υ1, υ2}satisfying
w0 −→w1w2
w0w1w2
n0w−1
2 ,
w1 −→w2
w0w1w2
n1,
w2 −→
w0w1w2
n2w
0.
4.3
Consider the universal covering p : R1→S1, which is defined bypx e2πxi. The set of all liftings offn0,n1,n2is
fn0,n1,n2,m|m0,±1,±2, . . .
, 4.4
wherefn0,n1,n2,m:R1→R1is defined by
fn0,n1,n2,mx λn0,n1,n2
x−xn0n1n21
x m, 4.5
Lemma 4.1. Let fn0,n1,n2 : S
1, V→S1, V be the map as above. Then SNf
n0,n1,n2;S
1 −V
|n0||n1||n2|.
Proof. Sincefn0,n1,n2 : S1, V→S1, Vdoes not send any componentpointof V into itself,
fn0,n1,n2,mx/x for any lifting fn0,n1,n2,m see 4.5 for definition of fn0,n1,n2 and any x ∈
p−1V. By definition, any fixed point class offonS1−V is surplus. Note that S1 −V has three components: ˙w
0, ˙w1, and ˙w2, which are, respectively, the interiors ofw0,w1, andw2. By a computation, any fixed point offn0,n1,n2on ˙w0has the form
exp2n0−2−2m/9n0−1πifor some integerm. Thus,
Fixfn0,n1,n2
∩w˙0
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
exp
2n0−2−2m 9n0−1
πi
|m−1,−2, . . . ,−n0
if n0>0,
∅ if n00,
exp
2n0−2−2m 9n0−1
πi
|m0,1,2, . . . ,−n0−1
if n0<0. 4.6
Note that the fixed point exp2n0−2−2m/9n0−1πiis the projection of the fixed point offn0,n1,n2,mon the component0,2/3ofp−1w˙0, that is,
exp
2n0−2−2m 9n0−1
πi
p
0,2 3
∩Fix fn0,n1,n2,m
. 4.7
By definition of fixed point class see Proposition 2.1, those |n0| fixed points lie in different fixed point classes offn0,n1,n2onS
1−V. It is obvious that each of these|n
0|fixed point classes has index sgndet1−Df∗ sgn1−9n0 −sgnn0, and hence is essential.
Similarly,fn0,n1,n2has|n1|essential fixed point classes on ˙w1, and has|n2|essential fixed
point classes on ˙w2. So, we are done.
Next two lemmas will show that thesefn0,n1,n2’s can be regarded as representations of
maps on the circle having periodic points with least period 3.
Lemma 4.2. Two mapsfn0,n1,n2andfn0,n1,n2are in the same relative homotopy-conjugacy class if and only ifn0, n1, n2andn0, n1, n2are the same up to a permutation.
Proof. It is a straight verification.
Lemma 4.3. Letfbe a self-map onS1. Iffhas a periodic orbitP with least period3, then as a relative
map, f : S1, P→S1, Pis relatively homotopy-conjugate to a mapf
n0,n1,n2 : S1, V→S1, Vfor some integersn0, n1, n2.
Proof. Let P {x0, x1, x2}. Since P is a periodic orbit with least period 3, we may assume that x1 fx0, x2 f2x0, and x0 f3x0. Note that the three points in P are distinct. There is a homeomorphism h : S1→S1 such that hxk vk, k 0,1,2. The relative map
h◦f◦h−1:S1, V→S1, Vis therefore relatively conjugate tof :S1, P→S1, P.
hencef1/3 2/3n0 for some integern0. Note thatR1 is simply connected.f|0,1/3 and
λn0,n1,n2|0,1/3are homotopic relative to{0,1/3}and form any integersn1andn2. Project this
homotopy down toS1, we will have a homotopy, keepingυ0andυ1fixed, fromh◦f◦h−1|w0to
fn0,n1,n2|w0for any integersn1andn2. Repeat this argument ath◦f◦h−1|w1andh◦f◦h−1|w2,
we will obtain an integer n1 satisfying f2/3 n0 n1 1, and an integer n2 satisfying
f1 n
0n1n24/3. Thus,f|0,1is homotopic, relative{0,1/3,2/3,1}, tofn0,n1,n2,0|0,1.
Project down toS1, it follows thath◦f◦h−1is relatively homotopic tof
n0,n1,n2, and therefore
f :S1, P→S1, Pis relatively homotopy-conjugate tof
n0,n1,n2:S1, V→S1, V.
We restate a result of L. Block as follows.
Lemma 4.4see12, Theorem A. Letfbe a self map onS1. Suppose thatfhas a fixed point and a
periodic point with least periodnn >1. Then one of the following holds:
if has a periodic point with least periodmfor everym > n,
iif has a periodic point with least period m for every m satisfying n>sm, where >s is
Sharkovskii’s order of natural number set given by
3>s5>s7>s · · ·>s2·3>s2·5>s2·7>s· · · >s8>s4>s2>s1. 4.8
Our main result is the following theorem.
Theorem 4.5. Let f be a self map on S1 having a periodic orbit P with least period 3. Then f has
periodic points of each least period except for 2 if it is not relatively homotopy-conjugate to f0,0,0 :
S1, V→S1, V, which is the standard 120 degree rotation.
Proof. ByLemma 4.3, f : S1, P→S1, Pis relatively homotopy-conjugate to a relative map
fn0,n1,n2:S
1, V→S1, V. From Lemmas3.2and4.1, we have that SNf;S1−P |n
0||n1| |n2|. Sincen0, n1, n2/ 0,0,0, SNf;S1−P>0. It follows thatf has a fixed point onS1−P. By usingLemma 4.4in the casen 3,f has periodic points with each least period except for least period 2.
According to relative homotopy-conjugacy classes, period 3 on the circle almost forces all the other periods with only one exception. Roughly speaking, the statement that period 3 implies every period is almost true for maps on the circle. In some sense, our results cannot be improved because the mapf−1,−1,−1has no periodic points with least period 2.
Our statements here give more information about the coexistence of periodic points on the circle, comparing with the results in4,5,12, and so on. The relative homotopy-conjugacy
classes refine the homotopy classes, indicated by degree, on maps on the circle, because of the following.
Proposition 4.6. The degree offn0,n1,n2isn0n1n21.
5. Periodic orbits with larger periods
Finally, we illustrate some examples to show what will happen if the least period of a given periodic orbit is larger than 5.
Example 5.1. Letmbe an integer withm≥5, mapgm :S1→S1is defined by
gm eθi
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ exp
3θ4π
m
i
if 0≤θ≤ 2π
m,
exp
−2θ 14π
m
i
if 2π
m ≤θ≤
4π
m,
exp
θ2mπ
i
if 4π
m ≤θ≤
6π
m,
exp
2θ−4π
m
i
if 6π
m ≤θ≤
8π
m,
exp
θ4mπ
i
if 8π
m ≤θ <2π,
5.1
and mapgm :S1→S1is defined by
g m eθi ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ exp
2θ 4π
m
i
if 0≤θ≤ 2π
m,
exp
−θ 10mπ
i
if2π
m ≤θ≤
4π
m,
exp
2θ− 2π
m
i
if4π
m ≤θ≤
6π
m,
exp
θ 4mπ
i
if6π
m ≤θ <2π.
5.2
Ifmis odd and larger than 5,gm has a periodic orbit with least periodm:
Ve0i, e4π/mi, e6π/mi, e8π/mi, e12π/mi, . . . , e2m−1π/mi,
e2π/mi, e10π/mi, e14π/mi, . . . , e2m−2π/mi. 5.3
It is evident thatgm : S1, V→S1, Vis not relatively homotopy-conjugate to any standard rotation, but has no fixed point.
Ifmis even and larger than 4,gm has a periodic orbit with least periodm:
Ve0i, e4π/mi, e6π/mi, e10π/mi, e14π/mi, . . . , e2m−1π/mi,
e2π/mi, e8π/mi, e12π/mi, . . . , e2m−2π/mi. 5.4
It is evident thatgm :S1, V→S1, Vis not relatively homotopy-conjugate to any standard rotation, but has no fixed point.
This example implies that except for nonrotation condition, more hypotheses are neces-sary to force a fixed point on the circle when we are given a periodic orbit with a larger least period.
Recall from the proof ofTheorem 4.5that the existence of a fixed point is the key point to have periodic points of other least periods, that is, to applyLemma 4.4. Thus,Example 5.1
Acknowledgment
This work is supported by NSF of China10771143and a BMEC grantKZ 200710025012.
References
1T. Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975.
2O. M. Sarkovskii, “Co-existence of cycles of a continuous mapping of the line into itself,”Ukrainski˘ı Matematicheski˘ıZhurnal, vol. 16, no. 1, pp. 61–71, 1964Russian.
3M. Misiurewicz, “Thirty years after Sharkovski˘ı’s theorem,”International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 5, no. 5, pp. 1275–1281, 1995.
4L. Block, J. Guckenheimer, M. Misiurewicz, and L. S. Young, “Periodic points and topological entropy of one-dimensional maps,” inGlobal Theory of Dynamical Systems. Proceedings of the International Con-ference held at Northwestern University, Evanston, Ill., 1979, vol. 819 ofLecture Notes in Mathematics, pp. 18–34, Springer, Berlin, Germany, 1980.
5L. Alsed`a, J. Llibre, and M. Misiurewicz,Combinatorial Dynamics and Entropy in Dimension One, vol. 5 ofAdvanced Series in Nonlinear Dynamics, World Scientific, River Edge, NJ, USA, 1993.
6M. Misiurewicz and Z. Nitecki, “Combinatorial patterns for maps of the interval,” Memoirs of the American Mathematical Society, vol. 94, no. 456, pp. 1–112, 1991.
7L. Alsed`a, J. Guaschi, J. Los, F. Ma ˜nosas, and P. Mumbr ´u, “Canonical representatives for patterns of tree maps,”Topology, vol. 36, no. 5, pp. 1123–1153, 1997.
8X. Zhao, “On minimal fixed point numbers of relative maps,”Topology and Its Applications, vol. 112, no. 1, pp. 41–70, 2001.
9H. Schirmer, “A survey of relative Nielsen fixed point theory,” inNielsen Theory and Dynamical Sys-tems (South Hadley, Mass, USA 1992), vol. 152 of Contemporary Mathematics, pp. 291–309, American Mathematical Society, Providence, RI, USA, 1993.
10X. Zhao, “Estimation of the number of fixed points on the complement,”Topology and Its Applications, vol. 37, no. 3, pp. 257–265, 1990.
11P. R. Heath and X. Zhao, “Nielsen numbers for based maps, and for noncompact spaces,”Topology and Its Applications, vol. 79, no. 2, pp. 101–119, 1997.
