15 problems

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 $\phi$ be an analytic self-map of the unit disk $\mathbb{D}$, and write $\phi_n=\phi^{\circ n}$ for its $n$-th iterate. Suppose that $\phi$ is type II parabolic, meaning that i…

Let $\mathscr{X}$ be a taut, acyclic complex manifold, meaning that $H_j(\mathscr{X},\mathbb{Z})=0$ for all $j\in\mathbb{N}$, and let…

Existence conjecture for polynomial flows with prescribed heteroclinic regions. For all $n,m\in\mathbb{N}$, there exists a complex polynomial $F_{n,m}$ of degree $n+m$ such that th…

Let $M$ be a Kähler manifold and let $f$ be a holomorphic partially hyperbolic diffeomorphism of $M$, with center distribution $E^c$. Kähler center-holomorphicity conjecture. The c…

Let $f$ be a holomorphic partially hyperbolic diffeomorphism, with center distribution $E^c$. Real-analyticity conjecture. The center distribution $E^c$ is real analytic. The conje…

Let $f$ be a holomorphic partially hyperbolic diffeomorphism on a complex $3$-manifold. Three-dimensional classification conjecture. Up to passing to a finite cover, $f$ is holomor…

Let $M$ be a compact complex manifold and let $f:M\to M$ be a holomorphic Anosov diffeomorphism. A complex infra-nilmanifold is a quotient of a complex nilmanifold by a finite grou…

Homogeneous-chambar singular-set conjecture. For every $k\geq 1$,

4-chambar classification conjecture. Up to affine conjugacy, every $4$-chambar on an open subset of $\mathbb{C}$ is one of the following types:

Let $\mathbb{D}$ be the open unit disk, and let $\varphi$ be an essentially linear fractional self-map of $\mathbb{D}$ with exactly one fixed point $w$ in $\overline{\mathbb{D}}$.…

Let $f\colon S^2\to S^2$ be a Thurston map with a hyperbolic orbifold that is realized by a rational map. Let $P_f$ denote its postcritical set, and let pullbacks be considered up…

Critical-orbit convergence conjecture. There exists a critical point $c_0$ of $f$ such that

Let $f$ be a Lattès map on ${\mathbb P^2}$. Lattès-map bifurcation accumulation conjecture. The map $f$ belongs to the closure of the interior of the bifurcation locus. This is pre…