8 problems

Triangle algorithm invariant-domain conjecture. The invariant domain for the Triangle algorithm in dimension $d=3$ is nonconvex and fractal. For the associated win-lose induction,…

Fractal invariant domain conjecture. The natural invariant domain for this algorithm possesses a fractal structure, in contrast to the simplex domains arising in the classical Tria…

Almost-everywhere entry conjecture. For Lebesgue-almost every point in $\triangle$, the region $U$ can be reached after finitely many iterations of the $\mathbb{R}^3(e,13,e)$ TRIP…

Hidden $\mathbb{R}^2$ structure conjecture. The $\mathbb{R}^3$ TRIP maps that have hidden $\mathbb{R}^2$ structure are precisely the maps

Full-measure conjecture. There is a subset of $\triangle$ of full Lebesgue measure on which condition (I') or condition (II') is satisfied.

Let $m$ range over the positive integers, and consider the Jacobi–Perron algorithm (JPA) expansion of … A positive integer $m$ is called eventually periodic when this expansion is…

Let $\Lambda_d=\{\bar x\in\mathbb{R}_+^d:x[1]\leq\cdots\leq x[d]\}$, and let $\mathcal{J}:\Lambda_d\to\Lambda_d$ be the Poincaré multidimensional algorithm defined by … where…

Finite-family Hermite matrix conjecture. There is a family of multidimensional continued fractions spanned by finitely many multidimensional continued fraction algorithms such that