8 problems

Let $\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}$, let $\overline{\mathbb{A}}=\mathbb{T}^{1}\times[0,1]$ be the closed annulus, and let…

Let $f:\overline{\mathbb{A}}\to\overline{\mathbb{A}}$ be a homeomorphism preserving Lebesgue measure, let $\widehat{f}$ be a lift, and let $\operatorname{Rot}(\widehat{f})$ be its…

Let $T$ be a lifted graph, let $F\in\mathcal{S}_1(T)$, and let $E$ be the finite exceptional set such that every rational number $p/q\in\operatorname{Rot}(F)\setminus E$ is realize…

Let $T$ be a lifted graph and let $F:T\to T$ be a map of degree one. Its rotation set is denoted by $\operatorname{Rot}(F)$. Whole rotation set conjecture. The whole rotation set…

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

Devil's-staircase conjecture for $\mathfrak{r}$. The graph of $\mathfrak{r}$ is a devil's staircase, with plateau regions corresponding to

For $(a,b) in (0,1)times mathbb{R}$, let $f{a,b}(x)=x+a+bsin(2pi x)$, and let its rotation interval be the interval of rotation numbers associated with the map. Fix an interval who…

Let $T$ be a lifted graph, let $F:T\to T$ be a map of degree one, and let $\operatorname{Rot}(F)$ denote its rotation set. Closedness conjecture. The whole rotation set…