100 problems

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…

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 $\mathcal{M}$ be the Mandelbrot set, and write … where $J_c$ is the Julia set of the quadratic map with parameter $c$. Star-shapedness conjecture. The set…

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…

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…

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…

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…

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…

The Fibonacci spectrum conjecture. The spectrum satisfies

Valence conjecture. For every non-Chebyshev quadratic polynomial, $J=J_1$ is true.

The Siegel-boundary conjecture. The source introduces the expectation that this situation should satisfy the stated conclusion, but the supplied span omits the conclusion itself.

Let $f(z)=z^2+c$ be a quadratic polynomial whose Julia set is full, and let $[-\beta,\beta]$ denote its spine. The critical value is $c$. Spine conjecture. The critical value belon…

Let $f$ be a polynomial with Julia set $J$, and suppose that $J$ is locally connected. Assume that almost every point of $J$ with respect to harmonic (Brolin) measure is the landin…

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…

Let $f(z)=z^2+c$ be a complex quadratic map satisfying the Collet–Eckmann condition, and let the basin of infinity be the set of points whose iterates under $f$ tend to infinity. A…

A polynomial $f$ is structurally stable if its dynamics are stable under small perturbations in the relevant parameter space. Density conjecture. The set of structurally stable pol…

Let $f_{\lambda}(z)=z^{2}|z|^{\gamma-2}+c$ be parametrized by $\lambda=(\gamma-2,c)$, and let $J_{\lambda}$ denote its Julia set. Consider parameters $\lambda$ with small…

Let $M$ be the Mandelbrot set, let $\bM$ denote its boundary, and for $P_c(z)=z^2+c$ let $J_c$ denote the Julia set. Zero-area conjecture. The sets $\bM$ and $J_c$ for every parame…

Let $T$ be a sufficiently hyperbolic polynomial and consider the branches of solutions of its renormalization equation. Contraction conjecture. All branches of solutions of the ren…

Let $f$ be a complex polynomial of degree $d\geqslant 2$, let $K_f$ be its filled-in Julia set, and let $J_f=\partial K_f$ be its Julia set. A component of $K_f$ is called critical…

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

Shishikura's connectivity conjecture. The Julia set of every Newton map of an entire function is connected.

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…