6 problems

Let $W$ be a set of words, let $F(W)$ denote the associated object, and let $(<_A,<_D)$ be the pair of orders used in the order condition. Order-condition characterization conjectu…

Let $L$) be a language on $k$ letters satisfying a symmetric order condition and having no connection. Its palindromic complexity is the number of palindromic factors of each lengt…

Let $W(\mathbb{Z}^d)$ be the group of permutations of $\mathbb{Z}^d$ with bounded range, acting naturally on $\mathbb{Z}^d$. Non-extensive amenability conjecture. When $d\geq 3$, t…

Lowest-accumulation-point conjecture. The upper BL spectrum of three-interval exchanges below $6$ is strictly included in twice the Lagrange spectrum below $3$; moreover, $6$ is th…

Relative unique-ergodicity conjecture. There is a neighborhood $\mathcal{U}$ of $0$ in $\mathrm{REL}$ such that $\mathbf{a}+\mathcal{U}\subset\mathbb{R}^d_+$ and…

The lines-and-curves conjecture. (Lines) If $\mathbf{a}$ is uniquely ergodic and $\ell$ is a line in $\mathbb{R}^d_+$ passing through $\mathbf{a}$, then there is a neighborhood…