39 problems

Let ${\cal C}$ be the connectedness locus and let ${\fam0\fourteenrm Per}_1(\lambda)$ denote the locus of maps having a fixed point with multiplier $\lambda$. Hausdorff-limit conje…

Consider the parameter space of complex Desboves maps with real coefficients. A cycle of attracting Herman rings is a periodic collection of Herman rings whose return dynamics is a…

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 $W$ be a hyperbolic component in the parameter space of exponential maps, and let $\mathcal{W}(W)$ denote its wake. A subwake of $W$ is a wake associated with a child component…

Consider a generic one-parameter family of maps. Density conjecture. Maps with exponential decay of correlations are Lebesgue density points of other maps with uniform exponential…

For each $n\geq 3$, consider the family $r_{n,a}$ in the $a$-plane and the parameters $a_k$ associated with the potential centers of hyperbolic components, where $k$ is odd and…

For the subfamily $r_{n,a}$, let $M_n(r_{n,a})$ denote its boundedness locus, let $\mathcal{S}_n$ be its finite-$n$ spine, let $\mathcal{S}_\infty$ be the limiting spine, and let…

Let $W$ be a region in a one-dimensional subfamily of the generalized McMullen parameter space, parameterized by $a$. Suppose one critical value $v_+(a)$ is topologically stable, w…

Let a one-dimensional subfamily of the bifurcation locus of the generalized McMullen family be given, with two critical orbits having independent behavior. Suppose one critical val…

Let $\mathcal{F}_b:=J_b\circ\operatorname{Cov}_0^Q$ be the family associated to a degree $d+1$ polynomial $Q$ with two critical points, one of which is simple, and an involution…

Let $F=F_{a,v}$ be a cubic polynomial in the superattracting period-$p$ parameter curve $4\mathcal{S}_p4$, written in the normal form … Assume that $F$ is a Misiurewicz map, meanin…

Tan Lei Similarity. Under iterated magnification, the connectedness locus around a Misiurewicz map $F\in{mathcal S}_p$ looks more and more like the filled Julia set of $F$ around t…

Let a Mandelbrot copy in ${\mathcal S}_p$ cut across the boundary between two escape regions at a parabolic point ${\mathfrak p}$ of period $q$. The associated dynamic configuratio…

Consider an A off-center configuration with the angles and identifications described immediately before the claim, so that Condition (2) is satisfied. A off-center configuration co…

Let ${\mathfrak p}$ be a parabolic point associated with parameters $p$ and $q$, and let the two side faces be the adjacent faces on either side of the relevant central face. A hyp…

Let ${\mathfrak p}$ be a parabolic endpoint of edges in ${\bf Tes}_q({\overline{\mathcal S}}_p)$, and let $H_{\mathfrak p}$ be its Type D hyperbolic component. Edge Monotonicity Co…

Let ${\mathfrak p}$ be a parabolic point of ray period $q$. A hyperbolic component is of Type D when its attracting orbit of period $q$ is disjoint from the marked periodic orbit o…

Let $\mathcal{F}_2$ denote the relevant parameter space and let $\operatorname{Cyc}_p(\mathcal{F}_2)$ be the marked cycle curve of period $p$. Connectivity conjecture. For $p\ne 3$…

Disconnectedness conjecture. The connectedness locus is disconnected:

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…

Periodically repelling class conjecture. Every periodically repelling combinatorial class of $\mathcal{C}(\mathcal{S})$ is a point.

Let $\mathcal M_{\infty}$ be the family under consideration, and consider shell components in dynamically natural slices. Shell components of period $1$ are known to be unbounded.…

Consider the family of unicritical polynomials of degree $d\geq2$ or the exponential family. A parameter fiber is trivial when it contains only the parameter itself; fibers are def…

Let ${\cal F}$ be one of the following families: unicritical polynomials of degree $d$, rational maps of degree $d$, the exponential family, or a family of entire maps with finitel…

Let $\Omega_{\Delta}$ denote the hyperbolic components arising in the parameter space under consideration, and let $S$ be the set of parameters where the conclusions of the relevan…