7 problems

Let $R$ be an $n\times n$ expanding integer matrix and let $0\in\mathcal D\subset\mathbb Z^n$ satisfy $\#\mathcal D=|\det(R)|$. Define $T(R,\mathcal D)$ to be the associated self-a…

Uniqueness conjecture. For any self-similar crystallographic or self-affine tile, the combinatorial structure of the neighbor automaton uniquely determines the iterated function sy…

Let $p,q,r$ be integers with absolute value at least $2$, let $s,t,a_i,b_j$ be real numbers, and define … and … Here $a_\infty$ and $b_\infty$ denote the limiting quantities used i…

Deng–Liu–Ngai conjecture. Under these hypotheses, the self-affine tile $T$ is homeomorphic to a $d$-dimensional ball if and only if its interior $T^\circ$ is connected. The conject…

3-ball conjecture. Every self-affine tile satisfying the conditions of Theorem is homeomorphic to a $3$-ball.

The handlebody conjecture. Every self-affine 3-manifold is homeomorphic to a handlebody.

The twindragons $\mathcal B$ and $\mathcal C$ are self-affine tiles in the plane. Ball-homeomorphism conjecture. The twindragons $\mathcal B$ and $\mathcal C$ are homeomorphic to a…