15 problems

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 $\mathbf{C}\to\mathbf{C}$ be a monic, centered polynomial of degree $d\geq 2$, and let $\mathcal{P}^{d-1}$ be the affine space of such maps. Let…

Let $\mathcal{H}$ be a hyperbolic component of disjoint type. Define its obstructed boundary by … Let $\partial^\infty_{\mathbb{Q}}\mathcal{H}$ be the subset of…

Let $d$ be the degree, let $\operatorname{Rat}_{d,\text{fm}}$ be the space of fixed-point-marked rational maps, and let…

Let $\mathcal H_d$ be the central hyperbolic component in the space of degree-$d$ rational maps. A rational map is geometrically finite if every critical point in its Julia set has…

Local connectivity conjecture. The boundary $\partial \mathcal H_d$ is locally connected at most maps $f\in \partial \mathcal H_d- \overline{\Phi}(\mathcal R)$. This extends the pr…

Milnor's Hausdorff-dimension conjecture. If $\mathcal H$ is not exceptional, then $\partial\mathcal H$ is a fractal set in the sense that

Milnor's non-local-connectivity conjecture. If the maps in $\mathcal H$ have an attracting cycle with two distinct free critical points in its attracting basin, then…

Non-local connectedness conjecture. For $d\geq 3$, $\partial \operatorname{\mathcal{H}}_d$ is not locally connected.

Milnor's conjecture. In the complementary case, the boundaries of corresponding hyperbolic components are semi-algebraic; that is, when the hypothesis of the fractal-boundary conje…

Milnor's conjecture. If there is a rational map having no indifferent cycles in the boundary of some hyperbolic component, then this boundary is a fractal set.

Let $\mathrm{PHD}_3$ denote the cubic Principal Hyperbolic Domain, and let the conditions known to be necessary for a polynomial to belong to $\mathrm{PHD}_3$ be given. PHD3 suffic…

Let $a\in\mathbb{D}^*$ be fixed and let $\lambda_0\in\mathbb{D}_{\mathcal{C}(a)}^*$ be such that $c_-(a,\lambda_0)$ belongs to an iterated preimage of $D_0(a,\lambda_0)$. A paramet…

A vein in the Mandelbrot set is an embedded arc joining a boundary parameter to the center of the main cardioid; along the real vein, an algorithm uses the binary expansions of bou…