45 problems

Universal multifractal formalism conjecture. One might conjecture that it always holds that

Trace singularity-spectrum conjecture. Theorem $$ holds for every $r\in\mathbb{R}$ simultaneously on a prevalent set.

Let $u$ be a probability measure on $mathbb{R}^d$, let $(omegak){k=1}^{infty}$ be independent random points distributed according to $nu$, and for $alpha0$ define … Write…

Frisch–Parisi conjecture. In that Baire functional space, any typical element $f$ obeys some multifractal formalism and satisfies

Iommi–Kiwi conjecture. The spectrum has at most two Lyapunov inflections.

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…

Let $\tilde{X}_{t}$ be the continuous version of the density at time $t$, and let $H(x)$ denote its optimal Hölder index at $x$. For $\eta$ in the relevant interval, let $D(\eta)$…

Second Legendre equation. For every $\alpha$ in the segment

Let $\theta_c$ be the critical parameter and let $mathfrak S(thetac)$ denote the corresponding random set of points of slow growth. Write $operatorname{dim{{rm H}}}$ for Hausdorf…

Prevalent singularity spectrum conjecture. For every $f\in\widetilde{b}^{\mu,\Psi}_{p,q}([0,1]^d)$,

Let $L_1$ and $L_2$ be jump functions, and consider their bivariate multifractal spectrum at values corresponding to bivariate level sets. Minimum-spectrum conjecture. Under mild h…

Let $F^\lambda$ be the parametrized von Koch function and let $d_{F^\lambda}$ denote its multifractal spectrum. Concavity conjecture. The multifractal spectrum $d_{F^\lambda}$ is c…

Spectrum-equality conjecture. Even though necessarily $\alpha_{x_0}(t)\neq\alpha_0(t)$, one has

Let $F_s$ be the function … with $s\in(1/2,1]$, and let $D_{F_s}^{(p)}$ denote its multifractal $p$-spectrum. A fractional derivation of order $1$ is understood as an operator that…

Fast-time dimension conjecture. There is a deterministic constant $c_0>0$ such that the Hausdorff dimension of the set of $a$-fast times is exactly

Let $D_1$ denote the information dimension of an eigenvector, characterized by … Let $\chi$ be the level compressibility. Bogomolny–Giraud conjecture. The information dimension and…

Anomalous-dimension symmetry conjecture. The anomalous dimensions should satisfy

Let $\{S_i\}_{i\in\mathcal{I}}$ be an IFS in $\R$ with transition graph $\mathcal{G}$. The weak separation conjecture. The IFS $\{S_i\}_{i\in\mathcal{I}}$ satisfies the weak separa…

Minimal loop-class conjecture. For every real $q$,

Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For each $q\neq 1$, let $b_\mu$, $\mathsf b_\mu$, $\mathsf B_\mu$, $B_\mu$, and $\Lambda_\mu$ denote the multifractal di…

Let $T_1,\dots,T_N$ be contracting non-singular $2\times 2$ upper triangular matrices, let $c_i,d_i$ be the diagonal entries of $T_i$, and let $\mu$ be a planar self-affine measure…

Equality of Hölder exponents. One has

Multifractal-change conjecture. The multifractal nature of the GST process will change under these less regularity assumptions on the ground state.

Finite-resolution correction conjecture. The corrected estimate recovers the structure function that would be obtained from infinite-resolution $p$-leaders:

Hölder exponent conjecture. The Hölder exponent of $\operatorname{Re}(\mathcal{B})$ satisfies