6 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…

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…

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 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…