16 problems

Let $F(x,y)=\left(y,\frac{a+y}{x}\right)$ with $a>0$ act on ${\cal U}=\{(x,y)\in\mathbb{R}^2:x>0,y>0\}$. The map has a first integral $V$, whose level curves are topological circle…

Let $\varepsilon_{\mathrm{c}}(\omega)$ be the critical function for an irrational rotation number $\omega$, and let $\mathcal{R}_{k}(\varepsilon)$ denote the residue of the perturb…

Let $\frac{m}{p}\in\mathopen{}\mathclose\bgroup\originalleft(0,\frac{1}{2}\aftergroup\egroup\originalright)$ be a rotation number, and let $\frac{m^-}{p^-}$ and $\frac{m^+}{p^+}$ d…

Let $M$ be a phase space, and let $F=(f_\beta)_{\beta\in B}$ and $G=(g_{\tilde{\beta}})_{\tilde{\beta}\in\tilde{B}}$ be families of diffeomorphisms of $M$. They are weakly topologi…

Let $\Gamma$ be a simple strictly convex curve with hyperbolic or parabolic monodromy, and let $\gamma$ be the corresponding rear track with rotation number $\rho(\gamma)$. Positiv…

Let $\Gamma$ be a smooth strictly convex front track whose bicycling monodromy is hyperbolic or parabolic, and let $\gamma$ be a corresponding closed rear track. Its integer-valued…

Let $f\colon\mathbb R/\mathbb Z\to\mathbb R/\mathbb Z$ be an analytic orientation-preserving circle diffeomorphism, let $F$ be a lift of $f$ to the real line, and let…

Consider the Lyness map with parameter $\alpha>0$ and let $\rho^v_\alpha$ denote its rotation number as a function of the level parameter $v$. Zeeman's conjecture. The function…

Let $D$ be the domain under consideration, and let $\triangle P_1P_2P_3$ be a triangle in $D$. Write $\psi_{\triangle P_1P_2P_3}$ for its associated billiard map,…

A one-dimensional tiling space has a map $F$ arising in the setting above, with rotation class $\mu$ when that class exists. Denjoy-type conjecture. Some version of Denjoy's theore…

Let $F$ be a lift of a circle diffeomorphism and let $\tau(F+\omega)$ denote its complex rotation number for $\omega\in\mathbb H$. For a real parameter $\omega_0$, suppose that the…

Let $\lambda=a/b$ and $\delta=r/s$ be rational numbers with $0<\lambda,\delta<1$, and let $\gamma>1$. Theorem 2 concerns the rational rotation number $\rho_{\lambda,\delta}=p/q$ un…

Let $F$ be the free group generated by $a$ and $b$, and let $w\in [F,F]$. Commutator-subgroup rationality conjecture. Then … This is a rationality conjecture for the maximum of…

Let $w$ be an arbitrary word, and let $r,s\in\mathbb Q$. For real $r,s$, write $R(w,r-,s-)$ for the supremum of $\operatorname{rot}^\sim(w)$ over representations in which $a$ and…

Let $w$ be a positive word of the form … Suppose that $R(w,r,s)=p/q$, where $p/q$ is reduced. Generalized lock inequality. Then … For the linear representation with $a=R_r$ and…

For a piecewise isometric system with irrational rotation number, let the elliptic islands be the stable regions in phase space, and let the parameter vary over the system's parame…