27 problems

Let $f(x)\in\mathbb{Z}[x]$ be a monic expanding polynomial. A polynomial has the Height Reducing Property (HRP) if, writing $|f(0)|=c$, there is a polynomial $g(x)\in\mathbb{Z}[x]$…

Let … and define … Thus, $\Omega_2$ is the set of double zeros in $(0,1)$ of power series with coefficients in $\{-1,0,1\}$. Infinitely many components conjecture. The set…

Let $(Ai){i e i}$ be a completely reducible family in $operatorname{GL}d(mathbb{R})^{mathcal{I}}$, with $kin{0,ldots,d-1}$ and $sin(k,k+1)$. Let $p$ be the number of functions in t…

Typical-interior conjecture. If

Let $E\subseteq\mathbb{R}^d$ be a self-similar or self-affine set that is not contained in a hyperplane. It is Newhouse projectively thick if, for every invertible linear transform…

Let $F$ be a self-affine set, allowing overlaps among the defining affine pieces. Its upper and lower box dimensions are denoted by $\overline{\dim}_B F$ and $\underline{\dim}_B F$…

Let an self-affine attractor be the attractor of an affine iterated-function system, and let ${\boldsymbol s}$ denote its surface regularity. A parallelepiped is an affine image of…

Dimension conjecture. For all $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, the set $A_{\lambda,\mu}$ has dimension strictly greater than $1$.

Top-boundary conjecture. For all $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, the set $\partial_{top}(A_{\lambda,\mu})$ is the graph of a continuous, strictly increasing funct…

Let $F$ be one of the Bedford–McMullen carpets under discussion, and for $0\leq\theta\leq 1$ let $\underline{\dim}_{\theta}F$ and $\overline{\dim}_{\theta}F$ denote its lower and u…

Visibility conjecture. For almost all $e\in S^1$,

Companion-matrix tile conjecture. The companion matrix $M$ and the digits $D=\{\boldsymbol{0},{\boldsymbol e}\}$ generate a tile for every such expanding polynomial.

Bi-Lipschitz rigidity conjecture. If 2-attractors are not affinely similar, then they are not bi-Lipschitz equivalent.

Principal-projection box-dimension conjecture. Both projection box dimensions should exist:

Graph-directed self-affine box-dimension conjecture. The box dimension of $E$ exists, so that

Self-affine full-dimension conjecture. Then $T_1,\ldots,T_N$ are similitudes with respect to some inner product on $\mathbb{R}^d$.

Let $E_{\mathbf{A, v}} = \bigcup\limits_{i=1}^K A_i(E_{\mathbf{A, v}}) + v_i$ be a self-affine set, where $\mathbf{A} = (A_1, \ldots, A_K) \in GL_r(\mathbb{R})^K$,…

Invariant-measure attainment conjecture. If the Bernoulli dimension formula fails, then there exists an ergodic invariant measure $\mu$ on $\Lambda_\Phi$ such that

Bernoulli dimension conjecture. For every Barański sponge $\Lambda_\Phi$, or at least every Barański sponge satisfying the coordinate ordering condition, one has

Let $A$ be an expanding integer matrix whose characteristic polynomial is … and let $\mathcal{D}=\{0,1,\dots,m\}v$ with $m\geq1$ and $v\in\mathbb{R}^2$ such that $\{v,Av\}$ is line…

A uniformly random self-affine set is obtained as the attractor of a random iterated function system of affine contractions, with the randomness specified by the model under consid…

Let $A$ be an expanding integer matrix, and let the height reducing property (HRP) denote the property of its characteristic polynomial used to determine connectedness of the self-…

Let $A\in M_2(\mathbb{Z})$ be expanding with $|\det(A)|=3$, let $f$ be its characteristic polynomial, and let ${\mathcal{D}}=\{0,1,b\}v$ with $b>1$ and $v\in\mathbb{R}^2$ such that…

Local fibre structure conjecture. The local fibre structure is typical for general self-affine sets whose defining maps can involve different rotations and whose basic pieces can h…

Typical local structure conjecture. This is the typical local structure for large classes of self-affine sets, both with and without the open set condition.