8 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…

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

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 $[\boldsymbol{T},\boldsymbol{o}]$ 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…