23 problems

Let $(T^1M,f^t)$ be the geodesic flow on a compact rank $1$ surface of nonpositive curvature. For $\alpha\geq 0$, let $L(\alpha)$ be the level set of Lyapunov exponents, and write…

Let $K$ be a number field, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a rational map of degree $d\geq2$ defined over $K$, and let $\alpha\in{\mathbb P}^1(K)$. Minkowski-dimension exi…

Let $\Sigma_{\alpha,\lambda}$ be the spectrum of the Sturmian Hamiltonian with frequency $\alpha$ and coupling constant $\lambda$. Bounded-type non-differentiability conjecture. Th…

For each quadratic irrationality $\alpha$, let $\Sigma_{\alpha,\lambda}$ be the spectrum of the corresponding Sturmian Hamiltonian at coupling constant $\lambda$. Transverse inters…

Let $\alpha$ be a Sturmian frequency of bounded type, and let $\Sigma_{\alpha,\lambda}$ be the spectrum of the corresponding Hamiltonian at coupling constant $\lambda$. Bounded-typ…

Let $\alpha$ be the frequency of a Sturmian Hamiltonian, and let $\lambda$ denote its coupling constant. Non-differentiability conjecture. The Hausdorff dimension of the spectrum,…

Intermediate-dimension interpolation conjecture. There exists a monotone function $\alpha(s)\colon[0,1]\to[0,\infty)$ such that

Let $H_{\lambda}^{(3)}$ be the three-dimensional Fibonacci Hamiltonian, and let $\operatorname{Sp}(H_{\lambda}^{(3)})$ denote its spectrum. Fibonacci spectrum components conjecture…

Let $H_{\lambda}$ be the one-dimensional Fibonacci Hamiltonian with coupling $\lambda$, and let $H_{\lambda}^{(d)}$ be its $d$-dimensional Fibonacci Hamiltonian. Write…

Let $\operatorname{Sp}_{+}(\alpha,1)$ be the positive half of the critical almost Mathieu spectrum, and let $\operatorname{dim}_{\mathrm{H}}$ denote Hausdorff dimension and…

Let $\operatorname{Sp}_{+}(\alpha,\lambda)$ denote the positive half of the spectrum of the almost Mathieu operator, and let $\operatorname{dim}_{\mathrm{H}}$ denote Hausdorff dime…

Exact-overlap conjecture. For iterated function systems acting on $\mathbb{R}$, the only mechanism preventing equality in the similarity dimension inequality, the similarity dimens…

Let $f\colon \mathbb{T}\to[0,1]$ be a Borel function, and let $\mu_f^{+}$ be the associated stationary determinantal measure on the metric space $(\{0,1\}^{\mathbb{N}},d)$, where ……

For integers $m,n$ with $(m,n)\neq(1,1)$, let $f_{m,n}(k)=mn-\frac{k(m+n-k)mn}{(m+n)^2}-\left\{\frac{km}{m+n}\right\}\left\{\frac{kn}{m+n}\right\}$, where $\{x\}$ denotes the fract…

Let $q>q_{KL}$, where $q_{KL}$ is the Komornik–Loreti constant, and let $\mathcal W_q$ be the projection of the difference set between the symbolic univoque set and its approximati…

Let $[\boldsymbol{T},\boldsymbol{o}]$ be a unimodular eternal Galton–Watson tree. Suppose its offspring distribution is in the domain of attraction of an $\alpha$-stable distributi…

For $m,n$ positive integers, let $\operatorname{Sing}_{m,n}^*(\omega)$ denote the set of $m\times n$ matrices with Diophantine exponent at least $\omega$ that are not trivially sin…

Let $\{X(x):x\in\mathbb{R}^d\}$ be a multivariate $(E,D)$-operator-self-similar symmetric $\alpha$-stable random field, where $E$ and $D$ are the domain- and range-scaling operator…

Let $S^1$ be the circle of external angles, let $\theta\in S^1$, and let $B_{\scriptscriptstyle\mathrm{comb}}(\theta)$ denote the combinatorial biaccessibility dimension. For…

Let $K$ be the self-similar attractor of an iterated function system with contraction ratios $r_i$ and associated self-similar measure $\mu$, and assume the open set condition. For…

Let $\bar{\kappa}$ be the Coulomb-gas parameter, let $\Delta_{r,s}$ denote the conformal weights defined for arbitrary integer labels (including non-positive $r$ or $s$), and let…

Let $\bar{\kappa}$ be the Coulomb-gas parameter, let $\Delta_{r,s}$ denote the conformal weights defined for arbitrary integer labels (including non-positive $r$ or $s$), and let…

Let a complex tree be a scale-invariant network with tree structure, and let $D_b$ denote its box dimension and $D_w$ its walk exponent. The mean round trip time averaged over pair…