65 problems

Dimension formula conjecture. Under these assumptions,

Kaplan–Yorke conjecture. The quantity $d_{KY}$ coincides with the Hausdorff dimension of any attractor.

Berry's conjecture. The graphs of $\Re u(x,t)$, $\Im u(x,t)$, and $|u(x,t)|^2$ have fractal dimension $D=n+\frac{1}{2}$ at most all irrational times $t$.

Entropy-minimal symbolic characterization conjecture. There exists an entropy minimal subshift

Let a hyperbolic set have stable and unstable slices, and interpret “fractal dimension” as either Hausdorff dimension or upper box dimension. Hasselblatt–Schmeling conjecture. The…

Eden conjecture. The maximum in the formula for $d_L$ is attained on an equilibrium or periodic orbit, rather than on a chaotic orbit. The conjecture concerns when the global Lyapu…

Wilkinson–Austin's conjecture. For all frequencies $b1$,

Last's conjecture. In general, $\dim_B^+\Sigma_\lambda$ does not bound $\tilde\alpha^\pm_u(\lambda)$ from above.

Let $b(t)$, $t\in[0,1]$, be a two-dimensional Brownian motion with $b(0)=0$. Its frontier of Brownian motion (FBM) is the boundary of the unbounded connected component of the compl…

Let $\mu$ be the Erdős measure and let $\widetilde\mu$ be the associated two-dimensional Erdős measure described in Section 1. The dimension of $\mu$ is $\dim(\mu)=0.995713\dots$.…

In dynamical critical site percolation on the triangular lattice, call a time two-colour percolation time if there is both a white infinite cluster and a black infinite cluster at…

Pluriharmonic-measure dimension conjecture. The dimension of pluriharmonic measure of domains in $\mathbb{C}^n$ is at most

Euclidean harmonic-measure dimension conjecture. The dimension of harmonic measure in $\mathbb{R}^n$ is conjectured not to exceed

Oksendal's conjecture. The dimension of harmonic measure in $\mathbb{C}$ should never exceed $1$, although the Hausdorff dimension of its support can be as large as $2$.

Let $\gamma$ be the trace of chordal $\textrm{SLE}_\kappa$. SLE trace dimension conjecture. When $\kappa<8$, the Hausdorff dimension of $\gamma$ is almost surely … The source gives…

Let $K_1$ be the time-one hull of chordal $\textrm{SLE}_\kappa$ in the upper half-plane, and let $\partial K_1$ denote its outer boundary. Kenyon's outer-boundary dimension conject…

Let $E \subseteq \mathbb{R}^n$, with $n>1$, be a Besicovitch set, meaning that $E$ contains a unit line segment in every direction. Its upper Minkowski dimension is the upper Minko…

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…

Let $T\subseteq \mathbb{Y}$ be a subtree, and write … For $d>\frac{1}{2}$ and $\delta\in\left[\frac{1}{2},d\right]$, consider win-lose alternating-move games on $T$ with payoff set…

For $d\in\\{4,5\\}$, consider the loop soups $\mathcal{L}_{1/2}^{\mathbb{R}^d}$ and $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^d}$. In dimensions $d=4,5$, the continuum loops are m…

Let $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^3}$ be the loop soup on the metric graph of $\mathbb{Z}^3$, and let $\mathcal{L}_{1/2}^{\mathbb{R}^3}$ be the corresponding loop soup…

Let $\Gamma$ be a non-elementary hyperbolic group generated by a finite set $S$, and let $\nu$ be a probability measure on $S$ with $\operatorname{supp}(\nu)=S$. Consider the branc…

Fractional Hausdorff dimension conjecture. In a suitable topology, the limiting set has fractional Hausdorff dimension

Entropy-minimal subshift conjecture. There exists an entropy minimal subshift representing $A$ if and only if, for every arithmetic progression $P$,

Let the attractor be generated by a gradient system with a global Lyapunov function, and suppose that all equilibria are hyperbolic. Denote its Hausdorff dimension by…