6 problems

Let $S$ be a dilation surface that is not a translation surface, and let $\theta\in S^1$ be a direction. A directional foliation is Morse-Smale if it has finitely many closed leave…

The Morse-Smale directional-flow conjecture. There is an open dense subset $U$ of $\mathbb{S}^{1}$ such that the directional flow of $X$ along any direction of $U$ is Morse-Smale.

The density conjecture for hyperbolic closed geodesic directions. The set of directions of hyperbolic closed geodesics of $X$ forms a dense subset of $\mathbb{S}^{1}$.

The hyperbolic closed geodesic conjecture. Every strict dilation surface contains a hyperbolic closed geodesic.

Generic Morse–Smale conjecture. The collection of directions $m\in\mathbb{RP}^1$ for which the directional foliation of $X$ is Morse–Smale has full measure.

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…