Higher-dimensional Morse–Smale conjecture for dilation surfaces and affine interval exchanges

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 piecewise affine circle homeomorphisms. Morse–Smale conjecture. The vertical foliation of almost every dilation surface in $\mathcal{MD}_{g,n}$ is Morse-Smale, and almost every $k$-affine interval exchange transformation is Morse-Smale. In particular, almost every piecewise affine circle homeomorphism is Morse-Smale. This conjecture proposes the extension of the preceding theorem to higher-dimensional moduli spaces and relates the generic dynamics of dilation-surface foliations to that of piecewise affine interval exchanges and circle homeomorphisms.