76 problems

Let $X$ be an irreducible variety defined over an algebraically closed field $F$ of characteristic $0$, and let $\phi\colon X\dashrightarrow X$ be a dominant rational map. A domina…

For an individual rational function $f\in\operatorname{Rat}_d(k)$, let $h_{\operatorname{crit},\mathbb{Q}}$ be the critical height on the moduli space $\mathrm{M}_d$, and let…

Adelic Zariski dense orbit conjecture. The stated adelic-open-set conclusion holds.

Let $Y$ be the variety in the preceding construction and let $X=F_r(Y)$, with $r\geq 2$, where $\Psi:X\dashrightarrow X$ is the Voisin rational map. For $z\in CH_0(X)_{hom}$, the s…

Let $p$ be a normalized polynomial, let $n\geq 2$, and let $K_{p,n}$ be the König root-finding map. Write $\Sigma p$ for the symmetry group associated with $p$ and $\Sigma K_{p,n}$…

Zariski Dense Orbit conjecture. If there is no non-constant rational function $g$ on $X$ satisfying $g\circ f=g$, then $\mathcal{Z}(f)\neq\emptyset$.

Zariski dense orbit conjecture. Either there exists $\alpha\in X(K)$ whose orbit under $\Phi$ is well-defined and Zariski dense in $X$, or there exists a non-constant rational func…

Let $(f_t)_{t ein\Lambda}$ be a non-isotrivial algebraic family of rational maps with degree $d\geq 2$, parametrized by an algebraic curve $\Lambda$ over $\mathbb C$. A rational ma…

The $4d-1$ bound conjecture. One can take

Let $f$ be a rational map. It is structurally stable if, within a holomorphic family containing it, all sufficiently nearby maps are topologically conjugate to it; it is hyperbolic…

Let $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ be a rational map of degree $d\geqslant 2$, and let $\mathrm{Rat}_{d}$ denote the space of Möbius equivalence classes of ration…

Let $f$ be a rational map, and let its algebraic entropy be the exponential growth rate of the degrees of its iterates. Bellon–Viallet conjecture. The algebraic entropy of every ra…

Let $k$ be a number field, let $S$ be a finite set of places of $k$ containing the archimedean places, and let ${\mathcal O}_{k,S}$ be the ring of $S$-integers. Let $R(x)\in k(x)$…

Let $\f:\mathbb{P}^2\dashrightarrow\mathbb{P}^2$ be a rational map, and let $\lambda_1=\lim_{n\to\infty}\operatorname{deg}(\f^n)^{1/n}$ be its first dynamical degree. The algebraic…

Let $\f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ be a rational map of topological degree $e$, and let $\lambda_1=\lim_{n\to\infty}\operatorname{deg}(\f^n)^{1/n}$ be its first dy…

Let $f:X \longrightarrow B$ be a proper morphism of integral varieties whose general fiber is an integral variety of general type. For a sufficiently large integer $n$, consider th…

Let $S$ be a very general K3 surface. A dominant rational map from the product of two smooth projective curves to $S$ is a rational map $C\times D\dashrightarrow S$ whose image is…

Conjecture. Every codimension-one foliation on $X$ either admits a singular transversely projective structure, or is the pull-back, by a rational map, of a foliation on a complex s…

Ji–Xie conjecture. If $f$ and $g$ have the same multiplier spectrum, then one of the following holds: $f$ and $g$ are Lattès maps, or $f\sim g$.

Generic length-spectrum injectivity conjecture. There is a Zariski closed proper subset $E_d\subset {\mathcal M}_d$ defined over $\mathbb R$ such that, for every $[f]\notin E_d$, i…

Small-period multiplier-spectrum injectivity conjecture. For every $d\geq 2$, the morphism

Multiplier-spectrum injectivity conjecture. If $\tau_d(f)=\tau_d(g)$, then either

Length-spectrum rigidity conjecture. If $f$ and $g$ have the same length spectrum, then

Uniform critical-orbit comparison conjecture. There exists a uniform constant $C_{}>0$ such that, for all $k\geq 0$,

Let $\mathcal{H}$ be a hyperbolic component of disjoint type. Define its obstructed boundary by … Let $\partial^\infty_{\mathbb{Q}}\mathcal{H}$ be the subset of…