46 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…

Computable-modulus conjecture. The function $f$ has a computable modulus of continuity.

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 ${\mathcal B}({\mathcal C})$…

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

Let $W$ be a region in a one-dimensional subfamily of the generalized McMullen parameter space, parameterized by $a$. Suppose one critical value $v_+(a)$ is topologically stable, w…

Let a one-dimensional subfamily of the bifurcation locus of the generalized McMullen family be given, with two critical orbits having independent behavior. Suppose one critical val…

Let $P$ be a polynomial having a parabolic cycle, and let $J(P)$ denote its Julia set. A small perturbation of $P$ should convert the immediate basin of the parabolic cycle into an…

Tan Lei Similarity. Under iterated magnification, the connectedness locus around a Misiurewicz map $F\in{\mathcal S}_p$ looks more and more like the filled Julia set of $F$ around t…

Let $g_a$ be the family described above, let $C_{p,2}$ denote the set of period-$p$ itineraries, and let … be the indicated partitions of the Julia sets for $a<-1$ and $a=0$, respe…

Let $f$ be a rational map, let $J$ be its Julia set, and let an invariant line field on $J$ mean a line field fixed by the pushforward operator … A rational map is double covered b…

Let $(f,P)$ be a marked rational map, with Julia set $J_f$, and let $G$ be a graph in the sphere. An $f$-invariant graph satisfies $f(G)=G$; the post-critical set is denoted by $P$…

Core local product structure conjecture. The set $\operatorname{Core}(\Lambda)$ has local product structure near any regular point, and is locally the product of a Jordan arc by a…

Let $f$colon ${\mathbb C}^2\to{\mathbb C}^2$ be a dissipative and hyperbolic automorphism, meaning that $|\operatorname{Jac} f|<1$, and suppose that $f$ has no attracting points. L…

Let $f$ be a hyperbolic rational map, and let $\phi$ be its virtual endomorphism. Write $N[\phi^n]$ for the corresponding complexity quantity at the $n$th iterate, and say that it…

Julia-set equality conjecture. For every rational map $\phi\in K(z)$,

Converse prisoner-set inclusion conjecture. If the radius $R$ is such that $\mathcal{P}(f)\subseteq\mathcal{P}(f_{c_1})$, then

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational function of degree at least $2$. Assume that the Julia set $J(f)$ is connected. Connected Julia set dichotomy conjec…

Nonexistence conjecture. The box-counting dimension of a typical connected Julia set does not exist.

Maximal area conjecture. The maximal area of $K_c$ is

Simple-dynamics conjectures. The following properties hold:

Alpha-limit conjecture. There is some non-empty open subset $A\subset f(\mathbb{R}\mathrm{P}^2)$ such that $\alpha_{N_f}(x)$ is equal to the set of non-regular points of the bounda…

Weak plainness conjecture. Then:

Let $f$ be a rational map, and let $J(f)$ denote its Julia set. An invariant line field on $J(f)$ is an invariant measurable field of unoriented lines defined on $J(f)$. NILF conje…

Let $f$ be a polynomial, an entire transcendental function with finitely many singular values, or a rational map that is not double covered by an integral torus endomorphism. An in…