7 problems

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 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 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…