26 problems

Hubbard's complexification conjecture. Regions in the real parameter space that appear unrelated may have a relation that becomes apparent after complexifying both the dynamical sp…

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…

Height-uniform boundedness conjecture. There exist $\epsilon>0$ and a constant $C$, depending on $K$, $B$, and $d$, such that, as $f$ varies over generalized Hénon maps of degree…

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

Uniform boundedness conjecture. As $f$ varies over generalized Hénon maps of degree $d$ over $K$, the quantity

Local foliation rigidity conjecture. The answer is no: there do not exist such $f$, $U$, and $\mathcal{F}$ for which $J^+\cap U$ is a union of leaves of the foliation.

Disconnectedness conjecture. The connectedness locus is disconnected:

Critical-locus partition conjecture. There exists a line containing ${\mathcal C}$ that partitions the plane so that the unstable periodic points are uniquely described by symbolic…

Braid type conjecture. The periodic orbits of Hénon maps are of the same topological type as the periodic orbits of the horseshoe.

Pruning front conjecture. The pruning front specifies the allowed symbolic dynamics, namely the union of all periodic points, completely: no orbits are pruned by mechanisms other t…

Let $\Sigma_{2}$ be the space of bi-infinite sequences on two symbols, let $\sigma$ be the two-sided shift on $\Sigma_{2}$, and let $\mathcal{HL}$ denote the horseshoe locus of Hén…

Cvitanović's pruned horseshoe conjecture. Every map $H_{a,b}$ in the Hénon family can be understood as a pruned horseshoe: after pruning some orbits of $F$, one obtains a map…

Milnor's conjecture. For such a parameter, some iterate of the dynamics of the Hénon map should have locally and approximately the same dynamics as the corresponding composed quadr…

Hénon rigidity conjecture. Period-doubling Hénon maps are foliated by codimension-$1$ rigidity classes determined by the average Jacobian $b$.

Let $K$ be a complete, locally compact non-Archimedean field with odd residue characteristic. For $(a,b)\in\mathcal H_\mathrm{II}^+$, let $\phi_{a,b}:K^2\to K^2$ be the associated…

Hénon's conjecture. The map $H$ has a non-uniformly hyperbolic attractor. This is a famous conjecture concerning the chaotic dynamics of the classical Hénon map; the supplied text…

Horseshoe-locus conjecture. All these parameters belong to the horseshoe locus, because there appear to be paths in $\mathcal{H}^{\mathbb{C}}$ joining them to points in $DN$.

Pruning Front Conjecture. Up to semiconjugacy, the real Hénon family is contained in $\mathcal{P}(F)$: for every such choice of real parameters $a,b$, the Hénon map $H_{a,b}$ belon…

Let $K$ be a number field, let $C$ be a curve over $K$, and put $F=K(C)$. Let $\varphi(x,y)$ be a Hénon map over $F$, and let $P,Q\in\mathbb{A}^2(F)$ have distinct orbits under…

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, and call it type-3 when it is neither type-1, meaning…