Stability of Rotation Pairs of Cycles for the Interval Maps

Abstract

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