Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 931484 | https://doi.org/10.1155/2011/931484

Taixiang Sun, Hongjian Xi, Hailan Liang, Qiuli He, Xiaofeng Peng, "Stability of Rotation Pairs of Cycles for the Interval Maps", Abstract and Applied Analysis, vol. 2011, Article ID 931484, 9 pages, 2011. https://doi.org/10.1155/2011/931484

Stability of Rotation Pairs of Cycles for the Interval Maps

Academic Editor: Stefan Siegmund
Received31 Oct 2010
Accepted18 Jan 2011
Published10 Mar 2011

Abstract

Let be the set of all continuous self-maps of the closed interval and has a cycle with rotation pair for any positive integer . In this paper, we prove that if , then , where is integer, odd, with coprime, and .

1. Introduction

Let be the set of all continuous self-maps of the closed interval . For any , we define the distance between and by Then becomes a metric space. For any subset of , we use to denote the interior of . A point is called a periodic point of with period if and for , and is called a cycle with period . Write , which is called the set of fixed points of . For any subset , we use and to denote the cardinal number of and the smallest closed subinterval of containing , respectively. Write if . For any positive integer , write has a cycle with period .

One of the remarkable results in one-dimensional dynamics is the Sharkovskii theorem. To state it, let us first introduce the Sharkovskii ordering for positive integers:

Theorem A (see [1]). For any positive integers and , if .
Block [2] studied stability of cycles in the theorem of Sarkovskii and obtained the following theorem.

Theorem B (see [2]). For any positive integers and , if .
Blokh [3] introduced the following ordering among all pairs of positive integers with .(1)If and or , then .(2)If , where are coprime, then if and only if .He also defined the rotation pair and the rotation number of cycles with period for the interval maps.

Definition 1.1 (see [3]). Let , be a cycle of with period , and . Then is called the rotation pair of and the rotation number of .
For any positive integer , write has a cycle with rotation pair .

Theorem C (see [3]). For any positive integers and , .
In this paper, we will study stability of rotation pairs of cycles for the interval maps. Our main result is the following theorem.

Theorem 1.2. If , then where is integer, odd, with , coprime, and .

2. Some Lemmas

In this section, we prove Theorem 1.2. To do this, we need the following definitions and lemmas.

Lemma 2.1 (see [4, Lemma 1.4]). Let . If are compact subintervals of with such that for , then there exists a point such that and for every .

Lemma 2.2. Let . If there are points , , and such that  (resp., ), then for any integers and with (resp. ), has a cycle with rotation pair satisfying for all and for all .

Proof. We only prove the case (the proof for the case is similar).
We may assume that , then for all . Choose . Then there exist points such that for every and . Let Then for and . By Lemma 2.1, there exists a cycle such that . Furthermore, can be renumbered so that with the desirable properties.

Lemma 2.3. Let ; then has a cycle with rotation pair such that for all and for all .

Proof. Let be a cycle of with rotation pair . We may assume that (the proof for the case is similar). Let ; then , and for each . We may also assume that there exists some such that otherwise; let which completes the proof of Lemma 2.3.
Let and and . Then for all . Let and and . Then . It follows from Lemma 2.2 that has a cycle such that with the desirable properties.

Definition 2.4 (see [4]). Let . A cycle of with odd period is called a cycle of Stefan type if or

Definition 2.5 (see [4, 5]). Let and be a cycle with period , where is odd and is an integer. For each and each , write . We call a strongly simple cycle if one of the following three conditions hold.(1)If , then either or and is a cycle of of Stefan type, that is, or (2)If and , then for each and each , and .(3)If and , then the following three conditions hold.(i)For each and each , , and .(ii)For each , is a cycle of of Stefan type.(iii) maps each monotonically onto another , with one exception.

Lemma 2.6 (see [4, 5]). If has a cycle with period , then has a strongly simple cycle with period .
Let be a cycle of with period . Then there is a unique map , which is called the linearization of , satisfying
(1) for all ,(2) is linear for all .
By Theorem 7.5 of [4], we know that if has a strongly simple cycle with rotation pair , then has also a strongly simple cycle with rotation pair .

Lemma 2.7. Let , where are coprime, , is odd, and is an integer. Then has a cycle with rotation pair satisfying
(1) if for and if for ;(2) is a strongly simple cycle of ;(3) cyclically permutes the sets ().

Proof. By Lemma 2.3, we may assume that is a cycle of with rotation pair satisfying Furthermore, we may assume that is the linearization of , , and be the unique fixed point of . Obviously, we have that for all and for all .
We may assume that (the proof for the case is similar). If , then it follows from Theorem 7.18 of [4] that Lemma 2.7 holds. Now we assume .
By Theorem C, has a cycle () with rotation pair satisfying and for all .
We can assume since otherwise there is nothing to prove. Furthermore, we may assume (the proof for the case is similar). Write ; then .
Claim 1. We may assume that there exists a positive number such that for all .Proof. Since , there exists a positive number such that for all or for all . If for all , then Claim 1 holds. Now we assume for all . Write since .
We claim that for all , . Indeed, if for some , then there exists a point such that ; thus , which implies . This is a contradiction.
We also claim . Indeed, if for some , then there exists a point such that since . Let ; then . Since , there exists a point such that , which implies and . This is a contradiction.
By using to replace , we know that Claim 1 holds. Claim 1 is proven.

Write . Let if and ; otherwise. Put .
Claim 2. are pairwise disjoint and .Proof. We first prove that are pairwise disjoint. Suppose that there exist and such that , then . Since , there exists a point such that , which implies . This is a contradiction.
Now we prove . Suppose that there exist some and such that , then , hence , which contradicts definition of . Claim 2 is proven.

By definition of , it follows that since otherwise we have , which is impossible.
If , then has a cycle with period . It follows from Claim 2 and Lemma 2.6 that has a cycle with rotation pair satisfying conditions (1), (2), and (3) of Lemma 2.7.
If , then and there exists a point such that . Thus . By Lemma 2.3 of [4], has a cycle of period 3 on . It follows from Claim 2, Theorem A, and Lemma 2.6 that has a cycle with rotation pair satisfying conditions (1), (2), and (3) of Lemma 2.7. Lemma 2.7 is proven.

3. Proof of Theorem 1.2

In this section, we will give the proof of Theorem 1.2.

Proof of Theorem 1.2. We may assume (the proof for the case is similar). Let . We wish to show that there exists a neighbourhood of in such that every has a cycle with rotation pair . The proof will be carried out in a number of stages.

Claim 3. If and , then there exists a neighourhood of in such that every has a cycle with rotation pair .

Proof. By Lemma 2.7, we know that has a cycle with rotation pair satisfying
(1) if for and if for ;(2) is a strongly simple cycle of ;(3) cyclically permutes the sets ().
For each , let denote the midpoint of the points in and . Then for each , we have either or Furthermore, the blocks can be renumbered so that for . Then Since for and , there exist points such that satisfying(1)for either or (2) for and and .Let and for all . Then for every and , we have if and only if , and . Put Then we have This yields a cycle such that for . it is easy to verify that the rotation pair of is . Claim 3 is proven.

Claim 4. If and , then there exists a neighourhood of in such that every has a cycle with rotation pair .

Proof. By Lemma 2.7, we know that has a cycle with rotation pair satisfying
(1)if and , then ; if and , then ;(2) is a strongly simple cycle of ;(3) cyclically permutes the sets ().
Since and , there exist points such that and for . Let and for all . Then for every , we have This yields a cycle such that for . it is easy to verify that the rotation pair of is . Claim 4 is proven.

Claim 5. If and , then there exists a neighourhood of in such that every has a cycle with rotation pair .

Proof. By Lemma 2.7, , has a cycle with rotation pair such that for all and for all .
Choose two integers with coprime such that . Without loss of generality, we can assume . Take . Put , then there exist points () such that with () and . Let and for all . Then for every , we have and .
Let and , , and .

Claim 6. for any .

Proof. Assume on the contrary that for some ; then there exists a point such that ; thus , which implies , a contradiction.

Claim 7. If , then has a cycle with rotation pair .

Proof. Indeed, if , let , then . Since and , we have . Let and and . Then . It follows from Lemma 2.2 that has a cycle with rotation pair .

In the following, we assume that .

Claim 8. or for any .

Proof. Assume on the contrary that for some ; then . Since , we have . Let ; then and there exists a point such that ; thus, , a contradiction.
By Claims 7 and 8, we know that has a cycle with rotation number . It follows from Theorem C that has a cycle with rotation pair , which completes the proof of Claim 5.

Theorem 1.2 now follows immediately from Claim 3, Claim 4, Claim 5, and Theorem C.

Acknowledgments

The project was supported by NSF of China (10861002), NSF of Guangxi (2010GXNSFA013106, 2011GXNSFA014781), SF of Education Department of Guangxi (200911MS212), and Innovation Project of Guangxi Graduate Education 105931003060.

References

  1. A. N. Sarkovskii, “Co-existence of cycles of a continuous mapping of the line into itself,” Ukrainskiĭ Matematicheskiĭ Zhurnal, vol. 16, pp. 61–71, 1964. View at: Google Scholar
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  3. A. M. Blokh, “Rotation numbers, twists and a Sharkovski\u\i-Misiurewicz-type ordering for patterns on the interval,” Ergodic Theory and Dynamical Systems, vol. 15, no. 1, pp. 1–14, 1995. View at: Publisher Site | Google Scholar
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  5. L. S. Block and W. A. Coppel, “Stratification of continuous maps of an interval,” Transactions of the American Mathematical Society, vol. 297, no. 2, pp. 587–604, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2011 Taixiang Sun et al. 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.


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