5 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…

Let $X$ be a finite-area translation surface with uniquely ergodic vertical flow, and let $s$ be a straight non-vertical slit joining two nonsingular points $P,Q\in X$. Gluing…

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…

Relative unique-ergodicity conjecture. There is a neighborhood $\mathcal{U}$ of $0$ in $\mathrm{REL}$ such that $\mathbf{a}+\mathcal{U}\subset\mathbb{R}^d_+$ and…

The lines-and-curves conjecture. (Lines) If $\mathbf{a}$ is uniquely ergodic and $\ell$ is a line in $\mathbb{R}^d_+$ passing through $\mathbf{a}$, then there is a neighborhood…