Stability of rotation pairs of cycles for the interval maps (Q535996)

Let \(C^0(I)\) denote the space of all continuous self-maps of the closed interval \(I\) with the supremum metric. Denote the interior of \(M \subseteq C^0(I)\) by \(\operatorname{int}M\). For \(n \in \mathbb Z^+\), let \[ P(n)=\{f\in C^0(I): f \text{ has a cycle with period } n \}. \] \textit{L. Block} [``Stability of periodic orbits in the theorem of Sarkovskii'', Proc. Am. Math. Soc. 81, 333--336 (1981; Zbl 0462.54029)] proved for any \(m,n \in \mathbb Z^+\) that if \(n\) implies \(m\) in the Sharkovskii ordering, then \(P(n) \subseteq \operatorname{int}P(m)\). For \(f \in C^0(I)\), \textit{A. M. Blokh} [``Rotation numbers, twists and a Sharkowskii-Misiurewicz-type ordering for patterns on the interval'', Ergodic Theory Dyn. Syst. 15, No. 1, 1--14 (1995; Zbl 0842.58018)] defined the rotation number and rotation pair for a cycle of \(f\) with period \(n>1\) and introduced an ordering \(\dashv\) among pairs of positive integers \((k,l)\) with \(k<l\). For any positive integers \(v>u\), let \[ P(u,v)=\{f \in C^0(I): f \text{ has a cycle with rotation pair } (u,v)\}. \] Blokh [loc. cit.] proved for any positive integers \(v>u\) and \(l>k\) that if \((u,v) \dashv (k,l)\), then \(P(u,v) \subseteq P(k,l)\). The main result is that if \((2^m ns, 2^m nt) \dashv (\gamma, \lambda)\) then \(P(2^m ns, 2^m nt)\subseteq \operatorname{int}P(\gamma, \lambda)\), where \(m \geq 0\) is an integer, \(n \geq 1\) is odd, \(1 \leq s<t\) with \(s\) and \(t\) coprime, and \(1 \leq \gamma < \lambda\).