133 problems

Gatzouras–Peres conjecture. There exists a unique ergodic $T$-invariant measure with the same Hausdorff dimension as $K$. Moreover, this measure is mixing for $T$ and is possibly m…

Let $\mathscr{T}$ be a hierarchical T-mesh, and let $\mathscr{C}$ be its CVR (cross-vertex relationship) graph. For a spline space with the highest order of smoothness, write…

For $n\geq 3$, let $b_n>0$ be a universal constant such that every domain $\Omega\subset\mathbb{R}^n$ satisfies $\dim w_\Omega\leq n-b_n$, where $w_\Omega$ denotes harmonic measure…

Let $K$ be the attractor of a self-similar iterated function system (IFS) $\{\varphi_i:i\in\Lambda\}$ on $\mathbb{R}$ that contains no exact overlaps. Let $\lambda_i$ be the contra…

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

Full dimension conjecture. For all $\lambda$ sufficiently close to $1$, one should have

A poset has a planar diagram if its cover graph can be drawn in the plane with every cover relation represented by a curve directed upwards. Its cover graph is the graph whose vert…

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.…

Let $\mathcal R$ be the Rauzy gasket, the attractor of the three projective maps $T_1,T_2,T_3$ acting on the standard $2$-simplex, induced by the matrices $N_1,N_2,N_3$ described a…

Let a self-similar set be generated by a finite iterated function system of similarities, and let its similarity dimension be the value defined by the usual similarity-dimension eq…

Let $f:\mathbb {P}^k\to\mathbb {P}^k$ be a holomorphic endomorphism of degree $d\ge 2$, and let $\mu$ be its equilibrium measure. Let … be the Lyapunov exponents of $\mu$. Binder–D…

Felsner–Reuter conjecture. In every diametral pair of linear extensions of $\mathcal{P}$, at least one of the two linear extensions reverses a critical pair of elements of…

Dimension-one conjecture. Every co-existentially closed continuum has Lebesgue covering dimension one.

First-type conjecture for Markov compacta. All Markov compacta are of the first type.

Product formula conjecture. For every compact metric space $Y$,

Let $X$ and $Y$ be uniform spaces, let $f:X\rightarrow Y$ be a cover, and suppose that $Y$ has uniform dimension at most $n$. Uniform dimension invariance conjecture. The spaces…

Let $X$ be a coverable uniform space with uniform dimension $u\dim X\leq n$, and let $\widetilde{X}$ denote its generalized universal cover. Uniform dimension conjecture. The unifo…

A space metrizable by a complete metric is considered, together with zero-dimensional metrizable groups and the property of being strongly zero-dimensional. Complete-metric embeddi…

Conjecture 5. There exist $\varepsilon>0$ and maps $f_i\colon\Delta^{n_i}\to\mathbb R^m$ such that, for every collection of maps $g_i\colon\Delta^{n_i}\to\mathbb R^m$ with each…

Let $f\colon X\to Y$ be a map of finite-dimensional metrizable compacta. For integers $0\leq t\leq d\leq T\leq m$, $d-t+1\leq q$, and $\dim f+d+1\leq m$, define…

Let $f\colon X\to Y$ be a map of finite-dimensional compacta, and let $C(X,\mathbb R^m)$ denote the space of continuous maps from $X$ to $\mathbb R^m$ with the uniform topology. Fo…

Let $f\colon X\to Y$ be a map of finite-dimensional metrizable compacta, and let $C(X,\mathbb R^m)$ denote the space of continuous maps from $X$ to $\mathbb R^m$ with the uniform t…

Resolution conjecture. There exist a compactum $Z$ with $\dim Z\leq n$ and a surjective map $r:Z\longrightarrow X$ such that

Let $X$ be a compactum, let $n\geq 2$ be an integer, and let $\mathcal G$ be a collection of abelian groups such that $\dim_G X\leq n$ for every $G\in\mathcal G$. Let…

Large-cardinal dimension-range conjecture. The appearance of large cardinal values in the functional representation of the dimension range of an espalier should be closely related…