13 problems

Let $f$ be a piecewise isometry of a region in Euclidean space, and let its attractor be the restriction domain under consideration. A dynamical system is uniquely ergodic when it…

Polynomial symbolic complexity conjecture. The symbolic dynamics of a PI has polynomial complexity: the number of different length-$n$ names of subcylinders appearing in the dynami…

Let $(\mathcal{Q}^{(n)},F|_{Y^{(n)}})$ be the piecewise isometry considered in the paper, with atoms given by the partition $\mathcal{Q}^{(n)}$. Finite convex refinement conjecture…

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…

Let $F$ be the family of translated cone exchanges parametrized by $0<\lambda<1$, $\alpha\in \mathbb{B}^{d+1}$, and $\sigma\in S(\{0,\ldots,d+1\})$ with $\sigma(0)=0$ and…

Let a piecewise isometry (PWI) be an area-preserving map with finite-perimeter atoms, and let its return plot encode the area and connectivity of the potentially ergodic region. Le…

Let $S$ be the hemispherical domain of a piecewise isometry (PWI), with cutting lines and singular set $E$; let $\bar{E}$ denote the closure of $E$, called the exceptional set. A f…

Universal-bound conjecture. There is a universal bound away from these pathological or degenerate cases.

Embedding conjecture. All minimal nearby IETs in $\mathcal{F}_4$ are continuously, or at least symbolically, embedded in $\Xi$.

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 $f_s$ be the tetrahedral twist on the surface $\tilde\Delta$ of a regular tetrahedron, with parameter $s\in[0,1)$, and let the associated tetrahedral PET family be obtained fro…

Cantor attractor conjecture. If $A$ is an interval, the topological attractor $X(w)$ is almost surely a Cantor set.

For a piecewise isometric system with irrational rotation number, let the elliptic islands be the stable regions in phase space, and let the parameter vary over the system's parame…