Period-set characterization conjecture for degree-one circle maps

Let ${\mathbb S}^1$ be the unit circle, let ${\mathcal L}_1(S)$ denote the class of maps under consideration, and let $F\in {\mathcal L}_1(S)$ have rotation interval

$${\operatorname{Rot}}_{\mathbb R}(F)=[c,d].$$

For $c\leq d$, define

$$M(c,d)=\{n\in\mathbb N: c<k/n<d\text{ for some integer }k\},$$

and, for $\rho\in\mathbb R$ and $S\subset\mathbb N$, define $\Lambda(\rho,S)=\varnothing$ when $\rho\notin\mathbb Q$, while $\Lambda(\rho,S)=\{nq:q\in S\}$ when $\rho=k/n$ in lowest terms. A tail of an ordering is a terminal subset of that ordering.

Period-set characterization conjecture. There exist sets $E_c,E_d\subset\mathbb N$, each a finite union of tails of the orderings $\leso{2}$ and $\leso{3}$, such that

$$\operatorname{Per}(F)=\Lambda(c,E_c)\cup M(c,d)\cup\Lambda(d,E_d).$$

Conversely, given $c,d\in\mathbb R$ with $c\leq d$ and nonempty sets $E_c,E_d\subset\mathbb N$, each a finite union of tails of the orderings $\leso{2}$ and $\leso{3}$, there exists a map $F\in {\mathcal L}_1(S)$ with ${\operatorname{Rot}}_{\mathbb R}(F)=[c,d]$ and

$$\operatorname{Per}(F)=\Lambda(c,E_c)\cup M(c,d)\cup\Lambda(d,E_d).$$

This conjecture seeks to extend Misiurewicz's characterization for degree-one circle maps from liftings on the real line to maps in ${\mathcal L}_1(S)$. The preceding theorem gives the analogous characterization for real liftings; the asserted description for maps on the specified space remains open in the supplied source.