12 problems

Let $\mathbf u$ be a binary generalized pseudostandard word, and let $\mathcal C_{\mathbf u}(n)$ denote its complexity, namely the number of factors of length $n$ in $\mathbf u$. T…

Word-complexity conjecture. The word-complexity is

Right-special-word conjecture. Based on the examples $\sigma=\sqrt{2}$ and $\sigma=\sqrt{3}$, the $(n-1)$-word that branches in two ways coincides with the unique $n$-word starting…

Unique branching conjecture. Exactly one $n$-word in $s(\gamma_\sigma)$ branches in two ways, while every other $n$-word has a unique extension to an $(n+1)$-word, for each…

Word-complexity conjecture. For such trajectories,

Word-complexity conjecture. Extrapolating from available data, one has

Let $w$ be an Arnoux–Rauzy infinite word. Its $2$-binomial complexity and subword complexity are functions assigning to each positive integer the number of equivalence classes of f…

Chen–Wen–Wu conjecture. For every $k\geqslant 3$, the function $\mathsf{b}_{\mathbf{t}_m}^{(k)}$ is ultimately periodic with period $m^k$.

Additive complexity conjecture. The additive complexity of a $k$-automatic sequence is a $k$-regular sequence.

Let $p(q)$ denote the word complexity of a subshift, namely the number of distinct words of length $q$, and suppose the subshift admits a weakly mixing probability measure. The qua…

Ferenczi's conjecture. The example described by Ferenczi, whose word complexity satisfies

Let $b\colon\mathbb{N}\rightarrow\mathbb{C}$ be a bounded sequence with subexponential word growth. Let $\lambda$ denote the Liouville function, and let…