44 problems

Keane's conjecture. Minimality implies unique ergodicity for interval exchange transformations.

Zorich conjecture. The Lyapunov spectrum of the Zorich cocycle is simple; equivalently, all the $\theta_i$ are distinct.

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 $\mathcal R$ be the Rauzy gasket, the attractor of the three projective maps $T_1,T_2,T_3$ acting on the standard $2$-simplex, induced by the matrices $N_1,N_2,N_3$ described a…

Marmi–Moussa–Yoccoz conjecture. The $\text{C}^1$ local conjugacy class of $T$ is a submanifold of codimension

Let ${\cal I}_n$ be the set of $n$-periodic points, let ${\cal I}'_n$ be the subset selected in the preceding construction, let $\Xi_m$ be the set of points considered when cycles…

Minimal interval exchange unique ergodicity conjecture. Every minimal interval exchange transformation is uniquely ergodic.

For a perturbed family of interval exchange maps, persistence of invariant curves leads formally, as $\varepsilon\to0$, to a cohomological equation over the limiting interval excha…

A rotational invariant curve is a continuous curve $\gamma$ homotopic to the line $y=0$ and invariant under the map, so that $T(\gamma)=\gamma$. For an unperturbed family of interv…

Consider a perturbed family of symmetric $d$--interval exchange maps and call a continuous curve homotopic to the line $y=0$ that is invariant under the map a rotational invariant…

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…

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 $T$ be a self-similar interval exchange transformation, let $\omega$ be a log-slope vector of unstable type, and let $\nu_\omega$ be any $\phi_\omega$-conformal measure for $T$…

Let $f$ be an affine interval exchange transformation semi-conjugated to a typical interval exchange transformation, meaning one arising from a full-measure set of parameters, and…

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 an IET-embedding be an embedding of an interval exchange transformation into a piecewise isometry, and suppose the embedding is not composed of line segments or circle arcs. Ke…

Arnol'd's mixing conjecture. In this case, the flow will in general be mixing on the open ergodic component.

Ghazouani–Ulcigrai conjecture. The a priori bounds should hold for typical minimal $C^3$ generalized interval exchange transformations $f$ with $mathcal N(f)=0$, without any assump…

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…

Marmi–Moussa–Yoccoz conjecture. For almost every choice of initial interval exchange transformation $T_0$, the set of generalised interval exchange transformations that are…

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…

The incompatibility conjecture. These conditions are not compatible with $\mu(E_1)=1$, and are not compatible with unique ergodicity. It is also unknown whether these conditions ar…

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