57 problems

The Mandelbrot set $\mathcal{M}$ is the connectedness locus of the quadratic family $p_c(z)=z^2+c$. MLC conjecture. The Mandelbrot set $\mathcal{M}$ is locally connected. Local con…

Density of hyperbolicity conjecture. The set $\mathcal{H}$ is dense in $\mathcal{M}$.

Let $f_c(z):=z^2+c$ for $c\in\mathbb{C}$, and let $\mathcal{M}$ be the set of parameters $c\in\mathbb{C}$ for which the orbit of the critical point $z=0$ does not escape to infinit…

Let $\bm M^d$ denote the set of parameters $c$ such that the Julia set of $z\mapsto z^d+c$ is connected; in particular, $\bm M^2$ is the Mandelbrot set. Douady–Hubbard local connec…

Holomorphic removability conjecture. The boundary $\partial M$ is holomorphically removable, and this should be provable by cutting neighborhoods of the set into pieces and establi…

A hyperbolic component of the Mandelbrot set is a connected component of its hyperbolic interior; its period is the period of the attracting orbit throughout the component. Hyperbo…

Let $P_c(z)=z^2+c$ be a quadratic polynomial, and call $P_c$ hyperbolic if it has an attractive or super-attractive periodic point in the complex plane. Define … Let $\mathcal{M}$…

Let $M$ be the Mandelbrot set for the quadratic family $Q_c(z)=z^2+c$, consisting of those parameters $c$ for which the orbit of $0$ is bounded. A hyperbolic component is a connect…

Let $f_a(z)=z^2+a$ and $f_b(z)=z^2+b$ be quadratic polynomials with connected Julia sets, and let $M$ be the Mandelbrot set. Say that a map in $M$ belongs to the $t$-limb if it has…

Mandelbrot hyperbolicity conjecture. There are no queer components in the Mandelbrot set; equivalently, every connected component of its interior is hyperbolic.

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…

Consider the quadratic family in the chosen normal form, and let $Q$ be the set of parameters for which the corresponding filled Julia set contains an interior point. Let $M$ be th…

Let $M(K)$ be the set of parameters $c$ for which $0$ has bounded orbit under iteration of $x^2+c$, and let $\mathcal{H}(K)$ be the subset for which $0$ tends to a formally attract…

Let $\M_2$ be the Mandelbrot set, and let the space of quadratic topological polynomials be the monotone model of its boundary constructed from equivalence classes of diameters and…

Let $\operatorname{Mol}$ denote the main molecule in the quadratic family $z^2+cz$, and let $Q$ be the quadratic map used in the source for the standard Basilica Julia set. Molecul…

Let $\mathcal{M}$ be the Mandelbrot set, let $\mathcal{H}\subset\mathcal{M}$ be a hyperbolic component, and let $\mathcal{M}_{\mathcal{H}}$ be the associated small copy of…

Bullett–Penrose conjecture. The connectedness locus of this family of algebraic correspondences is homeomorphic to the Mandelbrot set.

Dynamical-type refinement conjecture. Homotopies from $\mathcal{L}$ to $\mathcal{M}$ should be refined to align with the dynamical classification of eigenvalues.

Homeomorphism conjecture. There exists a continuous bijection

A reformulation of the MLC conjecture. The complement of the Mandelbrot set in $\mathbb{C}$ satisfies the visibility property outside a totally disconnected subset of the boundary.

Pseudo-monodromy marker conjecture. The pseudo-monodromy is described by the collection of markers

Lipa's pseudo-monodromy marker conjecture. The monodromy action of $\rho(\gamma)$ is described by compositions of the markers

Let $n$ be a return time in the setting of Proposition, and let the proposition's stated condition characterize return times. Converse conjecture. The converse statement of Proposi…

A Feigenbaum map is an infinitely renormalizable quadratic-like map with bounded combinatorics, meaning that all renormalization periods are bounded by some $bar p$. The renormaliz…

Let $\mathbf{M}$ denote the Mandelbrot set, the connectedness locus in the parameter space of quadratic polynomials. MLC conjecture. The set $\mathbf{M}$ is locally connected. This…