16 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…

Let $M$ be an expanding $3\times 3$ integer matrix and let $T=T(M,\mathcal{D})$ be a $3$-dimensional self-affine tile with collinear digit set … for some…

Handlebody conjecture. Every three-dimensional self-affine tile that is a topological three-manifold is homeomorphic to a handlebody.

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

Existence problem. Does there exist a closed 3-manifold that is a nontrivial self-affine tile?

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

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

Let $A\in M_{2}(\mathbb{Z})$ be an expanding similarity with characteristic polynomial $f(x)=x^{2}+px+q$, where $|q|\geq 2$. Let $\mathcal D=\mathcal D(v,|q|)$ and…

The paper considers the theorem asserting the relevant number-system characterization for disk-like self-affine tiles. Extension conjecture. Theorem can be extended to non-disk-lik…

Let $\mathbf{r}\in\mathcal{E}_n=\{\mathbf{r}\in\mathbb{R}^n:\varrho(\mathbf{M}_\mathbf{r})<1\}$, where $\mathbf{M}_\mathbf{r}$ is the companion matrix of $\mathbf{r}$, and for…

Let $A$ be a dilation matrix and let $D$ be a centered canonical digit set for $A$, with attractor $T(A,D)$. A tile is called stretched when it has no $A$-invariant lattice of tran…

Let $A$ be an integer dilation matrix similar to a Jordan matrix $J\in M_m(\mathbb{Z})$, let $D_J$ be the centered canonical digit set for $J$, and choose an integer matrix $P$ giv…

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…