243 problems

Let ${\cal P}_0$ be the collection of chess squares in the level-zero puzzle for the Feigenbaum map, and let $\operatorname{diam}(P)$ denote the diameter of a chess square $P$. Uni…

Let $Y$ be a smooth projective variety and $f:Y\dashrightarrow Y$ be a rational dominating map. A smooth birational model is a smooth projective variety $\hat Y$ together with a bi…

Let $f(z,w)=(p(z),q(w))$ be the product of two polynomial maps of degree $d$ in $\mathbb{C}$. The non-density conjecture. If $p$ or $q$ belongs to the bifurcation locus in…

Repelling 45-lines conjecture. On the $45$-lines $h_{19}$ is repelling and, hence, $\{X=0\}$ resides in the Julia set $J_{h_{19}}$.

Let $f_{6}$ be the map under discussion on ${\mathbf{CP}^{3}}$, and let the $5$-point orbit denote the orbit of the $5$-points. The basins of an attracting orbit are the sets of po…

Let $h_{11}$ be the map under discussion, and let ${\mathcal R}$ be the ${\mathcal S}_{5}$-symmetric real projective subspace ${\mathbf{RP}^{3}}$ preserved by $h_{11}$. The Julia s…

Let $h_{11}$ be the map under discussion on ${\mathbf{CP}^{3}}$, and let the $20$-point orbit denote the orbit of the octahedral $20$-points. A basin is the set of points whose ite…

Let ${\cal F}$ be the holomorphic family, let $B$ be a disk in the set of quasiconformally stable parameters, and let $J_\lambda$ denote the Julia set of the corresponding map. An…

Let ${\cal F}$ be the holomorphic family defined over ${\bf C}-\{0\}$, and let J-stable maps be the maps whose Julia sets vary stably in this family. The Density Conjecture. The hy…

Equidistribution conjecture. For every generic $H$, one has

Let $f$ be a polynomial of degree $d$ with an indifferent fixed point at the origin. Let $c_i$ be the critical points of $f$, let $\theta$ be the rotation number at the origin, and…

Consider the parameter space of complex Desboves maps with real coefficients. A cycle of attracting Herman rings is a periodic collection of Herman rings whose return dynamics is a…

Let $f$ be a rational map with an invariant complex elliptic curve. A periodic point is repelling when the derivative of the corresponding return map is expanding. Repelling-point…

Let $f$ be a rational map of ${\mathbb P}^2({\mathbb C})$, let ${\mathcal C}\subset{\mathbb P}^2({\mathbb C})$ be an $f$-invariant elliptic curve, let ${\mathcal B}({\mathcal C})$…

A solenoid by holomorphic curves is a compact solenoid in the complex projective plane whose leaves are holomorphic curves. Holomorphic-solenoid conjecture. There exists a compact…

Let $f$ be a complex rational function, and let $J(f)$ denote its Julia set. A compact metrizable space is finitely self-similar if it arises from a finite self-similarity system,…

Let $f_1,f_2,\ldots$ be a sequence of automorphisms of $\mathbb{C}^k$. Assume that there exist constants $0<a<b<1$ such that, for every $n$ and every $z$ in the unit ball, one has…

Let $f$ be a hyperbolic rational map, let $Z(s)$ denote its dynamical zeta function, let \\{\mu_j\} be the zeros of $Z$ counted with multiplicity, and let $\delta$ be the dimensi…

Let $T(z)$) be an expanding polynomial of degree $d$ with a real Julia set $E$, where $E\subset[-\xi,\xi]$ and $T^{-1}:[-\xi,\xi]\to[-\xi,\xi]$. For $x\in[-\xi,\xi]$, define the Ja…

Let $P_{\rm MCheb}(z):=z^2-1.87450961730020085\ldots$ be the real quadratic polynomial constructed so that its first-return combinatorics imitate those of the Chebyshev polynomial…

Let $E_{\kappa}(z)=e^z+\kappa$ be an exponential map, let $J(E_{\kappa})$ denote its Julia set, and let $I(E_{\kappa})$ denote its escaping set. An order-preserving conjugacy betwe…

Let $W$ be a hyperbolic component in the parameter space of exponential maps, and let $\mathcal{W}(W)$ denote its wake. A subwake of $W$ is a wake associated with a child component…

Fatou-set conjecture. For $n\geq 5$, $F_g$ consists of the basins of attraction of the $p_m$-points.

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…