9 problems

Let $\mathcal H$ be the Hénon family $F_{a,b}(x,y)=(x^2+a-by,x)$ with $a,b\in\mathbb R$, and let $\partial_{\operatorname{chaos}}$ denote the boundary in $\mathcal H$ of the set of…

Let $f_{a_*} \in \mathcal Q$ be an infinitely renormalizable quadratic polynomial with bounded type combinatorics. For $a,b \in \mathbb R$, let $F_{a,b}(x,y)=(x^2+a-by,x)$ be a Hén…

Generic period-two conjecture. For all but finitely many $\delta\in\mathbb{Q}$, the $\mathbb{Q}$-rational periodic points of every such map, with $c\in\mathbb{Q}$, have period divi…

Period classification conjecture. Over $\mathbb{Q}$, the map has no point of period $N$ except possibly when

Disconnectedness conjecture. The connectedness locus is disconnected:

Generic hyperbolicity conjecture. Generically, Kupka–Smale complex Hénon maps with dominated splitting are hyperbolic.

Let $f:\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a continuous family and let $U \subset \mathbb{R} \times \mathbb{R}^n$ be such that the periodic orbits in $U$ are contai…

Let $\mathcal{H}^{\mathbb{C}}_0$ be the component of the complex Hénon hyperbolic locus containing $\mathrm{HOV}$, let $(a_0,c_0)\in\mathrm{DN}$ be a basepoint, and let…

Let $(a,c)\in\mathcal{H}^{\mathbb{C}}\cap\mathbb{R}^2$ be a real parameter in the hyperbolic locus of the complex Hénon map. Such a parameter is of type-1 if…