169 problems

In the space of rational maps of a fixed degree, a hyperbolic rational map is a rational map for which every critical orbit converges to an attracting cycle. Fatou's density conjec…

No-wandering-domains conjecture. Such a rational map has no wandering domains in $\mathbb{P}^1_{\mathbb{C}_p}$.

Let $f$ be a rational map, let $J(f)$ denote its Julia set, and let $\widehat{J}(f)$ be the closure of the grand orbit of the singular values and periodic points. The Teichmüller f…

A rational map is a holomorphic map $f \colon \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ of degree at least $2$; its multipliers are the derivatives associated with its…

Let $f$ be a rational map of degree at least two. A point of the Julia set $J(f)$ is buried if it does not lie on the boundary of any component of the Fatou set. Makienko's conject…

Let $f$ be a polynomial of degree at least two, and let $K$ be a wandering Julia component of $f$. Such a component is necessarily of full type. Branner–Hubbard conjecture. Every w…

A rational map is eventually stable when its iterated arithmetic behavior satisfies the eventual stability property described in the paper. The conjecture concerns rational maps de…

Let $f$ be a rational map of degree $d\geq 2$ on the Riemann sphere, and let $\mathcal{M}_f$ be its moduli space of quasiconformal conjugacy classes. A rational map is flexible Lat…

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 $\Sigma K_{p,n}$…

Let $\phi$ be the rational map under consideration, with $J_{\mathbb{C}_p}(\phi)$ its $\mathbb{C}_p$-Julia set and $J_{\overline{\mathbb{Q}}_p}(\phi)$ the subset associated with…

Let $\phi(z)\in\mathbb{Q}(z)$ be a degree-$2$ rational map, and suppose its automorphism group satisfies $\operatorname{Aut}(\phi)\cong C_2$. A rational point has exact period $N$…

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…

A rational map is considered in a hyperbolic component of the space of rational maps, and its Julia set is a Sierpiński carpet. McMullen's boundedness conjecture. The hyperbolic co…

McMullen's conjecture. If a hyperbolic rational map has Julia set homeomorphic to the Sierpiński carpet, then the corresponding hyperbolic component is bounded.

Let $f\colon \mathbb{P}^1\to\mathbb{P}^1$ be a rational map, and let its Julia set be the set supporting the chaotic dynamics of $f$. An invariant line field on the Julia set is a…

Let $N(p,q)$ be defined for $q\ge2$ by … where $\phi(q)$ is Euler's phi function. Recursion conjecture. The recursively computed values $N(p,q)$ are the actual numbers of solutions…

Consider the degree-$3$ bicritical rational map commuting with the antipodal involution $\alpha(z)=-1/\overline z$, shown in the source with parameters $X=-.235$ and $Y=.213$, and…

Let ${\cal C}$ be the connectedness locus and let ${\fam0\fourteenrm Per}_1(\lambda)$ denote the locus of maps having a fixed point with multiplier $\lambda$. Hausdorff-limit conje…

Let ${\cal M}$ be the moduli space of bicritical rational maps and let … denote the region of maps in the connectedness locus for which all $n+1$ fixed points are strictly repellin…

Let $f$ be a hyperbolic rational map whose Julia set is a Sierpinski carpet. McMullen's conjecture. The map $f$ lies in a bounded hyperbolic component of the moduli space of ration…

Density conjecture. The expanding maps form a dense subset of $\operatorname{Rat}_d$ and $\operatorname{Poly}_d$.

Let $E_1$ and $E_2$ be the functionals on smooth maps $F:S^2\to S^2$ given by … and … Restrict them to rational maps of degree $k$ from $S^2$ to itself, and let a minimizer mean a…

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 $R(x)\in k(x)$…

Sharp two-critical-point conjecture. The optimal constant $K_2$ replacing $1/2$ in the preceding theorem is