21 problems

Non-zero asymptotic excess conjecture. If $\Sigma_\Psi$ has non-zero asymptotic excess, then, for every admissible $\mathbf y$, the transformation $T_{\Psi,\mathbf y}$ either is no…

Measure-zero conjecture. The subset $M_{\pi,\mathscr R}$ has zero Lebesgue measure in $X_{\pi,\mathscr R}$.

Let $n\in\mathbb{Z}_{>0}$ be odd. Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the circle, and let $E_2:\mathbb{T}\to\mathbb{T}$ be the doubling map. For observables … consider measur…

Let $r\geqslant 2$, and let $\mathrm{ITM}(r)$ denote the space of interval translation maps on $r$ intervals. An interval translation map is of infinite type when, for its attracto…

Upper bound conjecture. A FIET can have at most

Let $G=(V,E,l)$ be a connected metric graph endowed with an interval exchange transformation $T$ partitioning $G$ into $n$ intervals with lengths $\ell_1, \ell_2, \ldots, \ell_n$ l…

Let two interval exchange systems be isogenous, and suppose one is minimal and self-induced. Then the other is also minimal and self-induced. Associate to such systems their transl…

Let $(X,\operatorname{\sigma})$ be an interval exchange system (IES) with an invariant decomposition into ergodic probability measures, and let its rational invariants consist of t…

Let $0<\alpha<1$ be irrational, let $T$ be the rotation by angle $\alpha$, viewed as a two-interval exchange with discontinuity $1-\alpha$, and let $T_f$ be the Veech extension fro…

Let $X_N$ be the Veech $N$-example surface, with directional flow $\phi_t^{\theta}$, and let $T_f$ and $T_{1-f}$ be the associated interval exchange transformations. Write the slop…

Let $\CETn$ denote the family of maps used in the source, and consider periodic trajectories of a map $F\in\CETn$. Winding Conjecture. The winding number of a periodic trajectory d…

3-IET primality conjecture. Almost every $3$-IET is prime.

Existence conjecture. For any $d\geq 3$, there is a minimal $d$-IET $(I,f)$ which admits a continuous, non-trivial embedding into a $d$-PWI.

Let $\mathbb{S}^1$ be the circle, let $n$ be any positive integer, and let $F\in\operatorname{CET}_n$ be an interval exchange transformation with flips whose flip parameter is…

Let $g$ and $n$ be fixed such that the moduli space of dilation surfaces $\mathcal{MD}_{g,n}$ is non-empty. For $k\geq 3$, consider $k$-affine interval exchange transformations and…

Let $I=[0,1]$, let $X$ be an infinite subset of $I$, and let $T$ be an interval exchange transformation (IET) of $I$. Quantitative density conjecture. For almost every IET $T$, … T…

Let $I=[0,1]$, let $X$ be an infinite subset of $I$, and let $T$ be an interval exchange transformation (IET) of $I$. Almost-everywhere density conjecture. For almost every IET $T$…

Let $F_t$ be the affine interval exchange transformation associated with the parameter $t$, and let $\Lambda_{\Gamma}\subset S^1$ be the Cantor set of remaining parameters, defined…

Centralizer rigidity conjecture. For every integer $n\geq 2$,

Rich-restriction instability conjecture. If $\boldsymbol{\boldsymbol{\test}}$ is rich with respect to $\boldsymbol{\boldsymbol{\test}}$ and $T_{\boldsymbol{\boldsymbol{\test}}}$ sa…

Let $\mathcal{E}$ be the topological group of continuous interval exchange actions, and let $F=\{f_t\}$ and $G=\{g_s\}$ be one-parameter subgroups of $\mathcal{E}$. Noncommuting on…