22 problems

Let $bbL$ be a family of iterated function systems satisfying the UOSC and Condition. Let $bbP$ be a random code-tree measure, and assume that $F_\tau$ is not almost deterministic.…

Bugeaud–Durand conjecture.

Levesley–Salp–Velani conjecture.

Super-exponential condensation conjecture. Any analytic IFS which has super-exponential condensation but no exact overlaps must be sub-conjugated to a self-similar IFS.

Let $C^r$ denote the self-similar fractal family defined in the paper, and let Minkowski measurability mean existence of a finite, positive Minkowski content. The conjecture for…

Let $K$ be a self-similar set in the plane, and call a direction $\pi\in G(2,1)$ exceptional for orthogonal projections of $K$ if the projection dimension equality does not hold. C…

Feng–Huang–Rao's conjecture. If, for some $i\in I$, $\log\alpha_i\notin\operatorname{span}_{\mathbb{Q}}\\{\log\beta_j\\}_{j\in J}$, then

Logarithmic commensurability conjecture. The contraction ratios are logarithmically commensurable, meaning

Let $\Phi$ be a self-similar iterated function system on $\mathbb R^2$ and let $K$ be its attractor. The Luo–Rao–Xiong local connectedness conjecture. If $\Phi$ satisfies the open…

Let $E,F\subset\mathbb{R}^d$ be two totally disconnected non-trivial self-similar sets, generated by iterated function systems $\Phi=\{\varphi_i\}_{i\in I}$ and…

Let ${\cal A}$ be a wild labyrinth pattern: its graph is connected, it has at least one vertical and one horizontal exit pair, and it satisfies the stated labyrinth-pattern connect…

Polygonal-attractor conjecture. Every, not necessarily convex, attractor in $\mathbb{R}^d$ which is a polygon is a parallelepiped.

Exact overlap conjecture. If there is a dimension drop for $\Phi$, then $\Phi$ has exact overlaps.

Let $\mathcal{S}=\left\{S_i\right\}_{i=1}^{m}$ be a self-similar iterated function system on $\mathbb{R}^d$, with attractor $\Lambda$, similarity dimension $s$, and symbolic space…

Infinite-length arcs conjecture. The supermixed labyrinth fractal $L_{\infty}$ generated by $\{\tilde{\mathcal A}_k\}_{k\ge 1}$ has the property that, between any two distinct poin…

Let $F$ and $E$ be totally disconnected self-similar sets in $\mathbb{R}^d$, generated by the iterated function systems $\Phi=\lbrace\phi_i\rbrace_{i=1}^l$ and…

Let $\Phi=\{\phi_i(x)=\alpha_i x+a_i\}_{i=1}^m$ and $\Psi=\{\psi_j(x)=\beta_jx+b_j\}_{j=1}^{\ell}$ be self-similar iterated function systems on $\mathbb R$, with attractors…

Let $G$ be the relevant group of similarities of $\mathbb{R}^{d}$, and let $\Phi\subseteq G$ be an affinely irreducible iterated function system with attractor…

Regular-polygon packing measure formula. The conjectured packing measure satisfies

Exact-overlaps conjecture. A strict inequality in the preceding box-dimension bound can occur only in the presence of exact overlaps.

Let ${\boldsymbol{\rho}}=(\rho_1,\ldots,\rho_m)$ be a contraction vector such that $\operatorname{rank}\langle {\boldsymbol{\rho}}\rangle=m$, and let…

Let $K\subset\mathbb{R}^{d}$ be a compact set, and let $\mathcal{M}(K)$ be the space of probability measures on $K$ with its weak topology. For a measure $\mu\in\mathcal{M}(K)$, wr…