69 problems

Let $\Phi = \{x\mapsto r_i x + a_i\}_{i\in {\mathcal I}}$ be a self-similar iterated function system on $\mathbb R$, with $|r_i|\in (0,1)$, and let $\Lambda_\Phi$ be its attractor.…

Six-point conjecture. If $\frac{2}{3}\leq\lambda<\lambda_0$, then

Non-basis conjecture. None of the collections of orthogonal exponentials for $\lambda=\frac34$ is an orthonormal basis for the Hilbert space.

Let $R$ be an expansive $d$ by $d$ matrix over $\b7bZ$, and set $S=R^T$ to be the transposed matrix. Let $B,L\subset\mathbb{R}^d$ have the same cardinality $N$, and assume that the…

Let $\mathcal S_\lambda$ be the attractor of the three-map iterated function system in the paper, and let $\omega_m$ denote the golden-gasket parameters. Interior and measure conje…

For $\lambda\in(0,1)$, let $\mathcal S_\lambda$ be the attractor of the three-map iterated function system described in the paper, and endow each map with probability $1/3$. Let…

Let $f_i(x)=\lambda_i x+b_i$ be an affine iterated function system on the real line, and suppose that $|\lambda_i|\geq 1$ for at least one $i\in\{1,\dots,m\}$. Let $(\Omega,\nu)$ b…

Espigulé–Juher–Saldaña's conjecture. Analogous holes exist in $\mathcal M_n$ for every $n>2$.

Let $b$ be the base and let $D_1,D_2\subseteq\{0,1,\ldots,b-1\}$ be digit sets, with $\#D_1$ and $\#D_2$ denoting their cardinalities. A pair $(D_1,D_2)$ is called degenerate or Ke…

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.

Exact overlaps conjecture. In the self-similar setting,

Regular-closedness conjecture. For $3\leq n<21$, the set $\mathcal{M}_n\setminus(\mathbb{R}\cup\mathcal{X}_n)$ is regular-closed. The conjecture concerns the remaining part of thes…

Let $\mathcal{M}_2$ be the connectedness locus associated with the collinear fractals for $n=2$. The non-real part is $\mathcal{M}_2\setminus\mathbb{R}$. Bandt's conjecture. The se…

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

Exact overlap conjecture. If

Exact overlaps conjecture. If $\Phi$ has no exact overlaps, then

Let $1/2<\rho<1$ be a scaling factor, and let $\{\alpha^i\}_{i\in I}$ be the reduced banned words, enumerated in nondecreasing order of length, with $I\subseteq\mathbb{N}$. For eac…

Let $S=\{(x,y,z):x,y,z\geq 0,\ x+y+z=1\}$ be the standard two-dimensional simplex, and let $R\subset S$ be the Rauzy gasket, the unique non-empty compact attractor of the iterated…

The symbolic subsequence conjecture. There always exists an increasing sequence of natural numbers $\{c_n(\mathbf{t})\}_n$, dictated by the symbolic structure of $\mathbf{t}$, such…

Let $(A,L)$ satisfy the requirements of the strictly contractive dual iterated function system, let $\phi_0\in\mathcal{S}_{\mathrm{c}}(A)$, and let $\phi_\pi$ be the self-similar s…

Let $(A,L)$ satisfy the requirements of the strictly contractive dual iterated function system, and let $\phi_0\in\mathcal{S}_{\mathrm{c}}(A)$. Purity and compact-support conjectur…

Reversibility conjecture. If $A$ is homeomorphic to a ball and $\mathbf{i\in \Sigma}_{top}$ is both reversible and disjunctive relative to the top, then $\Pi_{top}(\mathbf{i})$ is…

Let $f_1(x)=c_1x+d_1$ and $f_2(x)=c_2x+d_2$ be affine maps with $ci otin{0}$, $|ci|<1$, and $ci,diinmathbb{Q}$, having distinct fixed points. Let $C$ be the attractor of this itera…

Let $({\mathbb X},d)$ be a metric space, and call $[x,y]:=\{z\in{\mathbb X}:d(x,z)+d(z,y)=d(x,y)\}$ the segment with ends $x,y$. A set is metrically convex if it contains $[x,y]$ f…

Let $F_t$ be a one-parameter iterated function system satisfying properties (H1), (H2), and (H3) of Section 4. If $A_{\bullet}=A^{\bullet}$ for some upper transition attractor of…