## Abstract and Applied Analysis

### Stability of Rotation Pairs of Cycles for the Interval Maps

#### Abstract

Let ${C}^{0}(I)$ be the set of all continuous self-maps of the closed interval $\mathrm{I}$, and $\mathbf{P}(u,v)=\{f\in {C}^{0}(I):f$ has a cycle with rotation pair $(u,v)\}$ for any positive integer $v>u$. In this paper, we prove that if $({2}^{m}ns,{2}^{m}nt)\dashv(\gamma ,\lambda )$, then $\mathbf{P}({2}^{m}ns,{2}^{m}nt)\subset \text{int}\mathbf{P}(\gamma ,\lambda)$, where $m\ge 0$ is integer, $n\ge 1$ odd, $1\le s\lt t$ with $s,t$ coprime, and $1\le \gamma \lt \lambda$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2011 (2011), Article ID 931484, 9 pages.

Dates
First available in Project Euclid: 12 August 2011

https://projecteuclid.org/euclid.aaa/1313171121

Digital Object Identifier
doi:10.1155/2011/931484

Mathematical Reviews number (MathSciNet)
MR2773643

Zentralblatt MATH identifier
1209.33017

#### Citation

Sun, Taixiang; Xi, Hongjian; Liang, Hailan; He, Qiuli; Peng, Xiaofeng. Stability of Rotation Pairs of Cycles for the Interval Maps. Abstr. Appl. Anal. 2011 (2011), Article ID 931484, 9 pages. doi:10.1155/2011/931484. https://projecteuclid.org/euclid.aaa/1313171121

#### References

• 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.
• L. Block, “Stability of periodic orbits in the theorem of Šarkovskii,” Proceedings of the American Mathematical Society, vol. 81, no. 2, pp. 333–336, 1981.
• A. M. Blokh, “Rotation numbers, twists and a Sharkovskiĭ-Misiurewicz-type ordering for patterns on the interval,” Ergodic Theory and Dynamical Systems, vol. 15, no. 1, pp. 1–14, 1995.
• L. S. Block and W. A. Coppel, Dynamics in One Dimension, vol. 1513 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1992.
• 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.