12 problems

Let $k$ be a complete metrized field, let $X$ be a projective variety defined over $k$, and let $f\colon X\dashrightarrow X$ be a dominant rational map. For any complete metrized f…

Let $k$ be an algebraically closed complete non-Archimedean field, possibly of positive characteristic. Let $\mathcal{J}$ be the Julia set of a polynomial of degree at least $2$ th…

Uniform bounded-period conjecture. There exists an integer $N(d)\ge1$ such that every such map $f$ admits a rigid repelling periodic point of period at most $N(d)$.

Fix a rational map $R$ over $K$ of degree $d\geq 2$, let $\chi(R)$ be its Lyapunov exponent, let $\mathbb{H}_K$ denote the hyperbolic space, and let $\operatorname{RFix}(R^n)$ be t…

Fix a rational map $R$ over $K$ of degree $d\geq 2$, and for $n\in\mathbb{N}^*$ let $\operatorname{RFix}(R^n)$ be the set of repulsive fixed points of $R^n$. Let $\mu_R$ be a measu…

Fix a rational map $R$ over $K$ of degree $d\geq 2$. A fixed point $x$ is called repulsive if either $x\in \operatorname{\mathcal{C}}_R\cap \mathbb{H}_K$, or $x\in \mathbb{P}^1_K$…

Let $k$ be any complete non-Archimedean field, and let $X$ be a projective $k$-variety. A morphism $f\colon X\to X$ is polarized if there exists an ample line bundle $L\to X$ and a…

Lindahl's conjecture. The polynomial $f$ is linearizable if and only if $a_i=0$ for every $i\geq 2$ such that $p\nmid i$. This conjecture refines Lindahl's known examples of non-li…

Let $K$ be the non-archimedean field under consideration, let $f_x$ be a rational map, and let $J_x$ be its Julia set. Assume that $\operatorname{char}(K)=0$ and that $J_x$ contain…

Let $K$ be a complete, locally compact non-Archimedean field with odd residue characteristic. For $(a,b)\in\mathcal H_\mathrm{II}^+$, let $\phi_{a,b}:K^2\to K^2$ be the associated…

Let $f$ be an endomorphism of the $k$-analytic projective space, and let $F_{\mathrm{norm}}(f)$ and $F_{\mathrm{harm}}(f)$ denote its normal and harmonic Fatou sets, respectively.…

Herman's divergence conjecture. For such fields, the formal conjugacy usually diverges, even for polynomials of one variable.