3 problems

Finite-or-dense orbit conjecture. For every $x_0\in U$, the orbit $\mathcal O(f,x_0)$ is finite or dense in $U$.

Let $S$ be the topological disc consisting of four triangles from the preceding examples, and let $R\colon S\to S$ be its continuous, area-preserving, piecewise affine first-return…

Let $S$ be a polygon in ${\mathbb R}^2$, and let $R\colon S\to S$ be a continuous, area-preserving, piecewise affine map such that $R=id$ on $\partial S$ and $R$ has infinitely man…