23 problems

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…

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 $F=F_{a,v}$ 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…

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 $\mathcal{F}_2$ 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:

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 $\Lambda$ be a dynamically natural slice for a family of meromorphic maps $\{f_\lambda,\ \lambda \in \Lambda\}$ in $\mathcal{M}_\infty$ in the strict sense. Let…

Let ${\mathcal M}$ be the parameter set under consideration, and let $\partial {\mathcal M}$ denote its boundary. A hole is a connected component of the complement of…

Let ${\mathcal M}$ be the parameter set under consideration, and let $\partial {\mathcal M}$ denote its boundary. A point of $\partial {\mathcal M}$ is algebraic if it is algebraic…

Tan Lei's parameter-space conjecture. The fundamental part of the cubic Newton family is homeomorphic to the quotient of the union of those two subsets by the equivalence relation…

Let the centered unit disk in the parameter plane be the disk centered at $0$ of radius $1$, and let a flare be one of the sun-flare-like parameter sets arising in the coded-colori…

Let $c$ be a parameter in the parameter plane, let $K(c)$ be the associated filled-in set, and let a coded-coloring be the coloring procedure described in the source using symbolic…

Let $c$ be a parameter in the parameter plane, let $K(c)$ be the associated filled-in set, and let the polygonal locus denote the parameter-space locus discussed in the source. An…

Trivial-fibers conjecture. In the parameter spaces of quadratic polynomials and exponential maps, all fibers are trivial.