13 problems

Conjecture de pré-périodicité des composantes. Toutes les composantes de l'ensemble de Fatou sont pré-périodiques.

Pure point spectrum conjecture. If $d_\infty=N$ and this condition holds for every real $\lambda$, then

Let $f,g\in\mathbb{C}(z)\setminus\mathbb{C}$ be such that $\deg(f)\geq2$ or $\deg(g)\geq2$, and let $c\in\mathbb{C}(z)$. Define … Also let $\hat{S}_{f,g,c}^2$ denote the correspond…

Let … where $\{x\}=x-\left\lfloor x\right\rfloor$. For a rational number $x\geq 2$, define its order under $\mathcal{T}$ to be the least non-negative integer $n$ such that…

Let $f$ be a rational map, let $J$ be its Julia set, and let an invariant line field on $J$ mean a line field fixed by the pushforward operator … A rational map is double covered b…

Let $f$ be a rational map of the Riemann sphere, and fix its degree. A rational map is hyperbolic when every critical point is attracted to an attracting periodic cycle. Fatou's de…

Let $S$ be a finitely generated semigroup of rational maps, and say that a rank $2$ free subsemigroup is a free subsemigroup generated by two elements. An invariant probability mea…

Let $S$ be a semigroup of rational maps. An invariant probability measure is a probability measure invariant under every element of $S$, and a non-injective element is a map in $S$…

Let $K$ be a global field of characteristic $0$ or greater than $2$, and let $\phi(x)\in K(x)$ have degree $2$. Assume that $\phi$ is not post-critically finite and that $0$ is not…

Let $\phi(x)=b(x^2+1)/x$ with $b\in\mathbb{Q}$ and $b\notin\{0,\pm 1/2\}$. Let $T_\infty$ be the pre-image tree rooted at $0$, let $G_\infty(\phi)$ be its arboreal Galois group, an…

Let $K$ be a field, let $\phi\in K(x)$, and let $T_\infty$ be its pre-image tree rooted at $0$. Let $A(\phi)$ be the subgroup of $\operatorname{Aut}(T_\infty)$ generated by the act…

Let $f$ be the rational function considered in the paper. Let $\mathcal{A}$ be the alphabet whose symbols $I_i$ correspond to the intervals produced by $pa$ on the real axis, and l…

Hinkkanen–Martin's conjecture. The set $E(G)$ is Möbius equivalent to a line segment or a circle.