19 problems

Connected Bone Conjecture. Every bone $B_\pm^{\rm cub}({\bf o})$ for the cubic family is a simple arc.

Monotonicity conjecture. The topological entropy of a real cubic map depends monotonically on its parameters, in the sense that every locus of constant entropy in parameter space i…

Let $s$ be the growth number of the real cubic family, and consider a constant-growth-number locus in the parameter plane. Algebraic-integer conjecture. If this locus has interior…

Let $s(A,b)$ denote the growth number of the real cubic map $x\mapsto x^3-3Ax+b$, and fix a level set $\{(A,b):s(A,b)=s\}$. A map is hyperbolic when its critical orbits converge to…

Consider the classification of figures by dynamic type and topological shape, and let a totally non-hyperbolic unimodal case mean a unimodal combinatorial case that is totally non-…

Consider the $+-+$ bimodal region and the loci corresponding to constant topological entropy. Plus-minus-plus entropy-disconnection conjecture. In the $+-+$ bimodal region, the cor…

Let ${\bf h}(f)$ denote the topological entropy of a quadratic map $f$, and let a constant-entropy locus be a set of parameters in the unimodal region on which ${\bf h}$ is constan…

Let a minimal unimodal combinatorics be a minimal combinatorial type in the unimodal region, and call it unobstructed when it is realized by the corresponding quadratic dynamical s…

Let an NH-bone be a connected component in $\overline{\mathcal U}$ of the locus where the critical point $c_1$ is eventually periodic and repelling, with specified eventual period…

For each $d\geq 1$, define the Hermite polynomial by … Let $\mathscr{R}_d$ denote the set of real polynomials of degree $d$ having only real periodic points. Hermite polynomial con…

Codimension-one existence conjecture. These codimension-one hyperbolic maps exist on every isentrope, with trivial exceptions such as $h=\log 3$ for bimodal maps.

Hyperbolic-density conjecture. In the space of polynomials of degree $d>2$ with all critical points real, there are no isentropes of entropy

Milnor's connectedness conjecture. Every isentrope in this family is connected.

Weak plainness conjecture. Then:

Plainness conjecture. Every $p$ satisfying the hypotheses of Barna's theorem is a plain map.

Critical-points conjecture. Every critical point $c_n$, for $n>0$, is attracted to one of the cycles $\mathcal{A}_1$ or $\mathcal{A}_2$.

Empty-interior conjecture. The set $U_f^{\infty}$ has empty interior.

Stable-set conjecture. The function $f$ has no attracting cycle in the interval $[\mu_3,+\infty)$.

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…