10 problems

Generator-number conjecture. The highest possible level of an orbit is $n+1$.

Smoothness conjecture. Orbits of points in Case 4 will consist of a countable number of copies of an interval $I$, the union of a number of subintervals $J_n$, in each of which the…

Fuchsianity conjecture. The group $G$ is a Fuchsian group, equivalently, a convergence subgroup of $\operatorname{Homeo^+}(S^1)$.

Let $\Sigma$ be the nondecreasing lift associated with the factorization map, and write $d\Sigma=\Sigma' d\theta$ when its measure is absolutely continuous with respect to Lebesgue…

Fix a permutation $p$ and write $\sigma(p,w)(e^{i\theta})=e^{i\Sigma(\theta)}$, where $\Sigma$ is the associated nondecreasing lift. Let…

Let $\operatorname{Homeo}(S^1)$ be the group of homeomorphisms of the circle, and let $PSU(1,1)^{(n)}$ be the $n$-fold covering subgroup induced by the projective action of…

Fix a permutation $p$ of the natural numbers. For $s\geq \tfrac12$, let $w^s$ be the space of sequences $w=(w_n)$ satisfying … and let $\prod_{n=1}^{\infty}\Delta$ denote the produ…

Fix a permutation $p$ of the natural numbers, and let $\sigma_N$ be the associated finite factorization maps with limit $\sigma(p,w)$, where $w=(w_n)$ belongs to the product of uni…

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…