Abstract and Applied Analysis
Volume 2011 (2011), Article ID 931484, 9 pages
doi:10.1155/2011/931484
Research Article

Stability of Rotation Pairs of Cycles for the Interval Maps

Taixiang Sun,1 Hongjian Xi,2 Hailan Liang,1 Qiuli He,1 and Xiaofeng Peng1

1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
2Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Received 31 October 2010; Accepted 18 January 2011

Academic Editor: Stefan Siegmund

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.

Abstract

Let 𝐶 0 ( 𝐼 ) be the set of all continuous self-maps of the closed interval 𝐼 , and 𝐏 ( 𝑢 , 𝑣 ) = { 𝑓 ∈ 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 𝑛 𝑠 , 2 𝑚 𝑛 𝑡 ) ⊣ ( 𝛾 , 𝜆 ) , then 𝐏 ( 2 𝑚 𝑛 𝑠 , 2 𝑚 𝑛 𝑡 ) ⊂ i n t 𝐏 ( 𝛾 , 𝜆 ) , where 𝑚 ≥ 0 is integer, 𝑛 ≥ 1 odd, 1 ≤ 𝑠 < 𝑡 with 𝑠 , 𝑡 coprime, and 1 ≤ 𝛾 < 𝜆 .