Mathematics Problem Archive

For an oval $\gamma$, the area spectrum of its outer billiard is the set of areas of the circumscribed polygons formed by periodic outer-billiard traj...

Can polynomial tunings be constructed by quasiconformal surgery?...

For a fixed polynomial $P_1$, does the polynomial obtained by tuning $P_1$ with $P_2$ vary continuously with $P_2$?...

Among polynomials $P_1$ of degree greater than two with a superstable orbit of fixed period, does the tuning with a fixed $P_2$ vary continuously with...

Let $P_{1,k}$ have a superstable orbit whose period tends to infinity and suppose $P_{1,k}\to P_{1,\infty}$. Do the tunings with a fixed polynomial $P...

When does the topological intertwining construction for two polynomial dynamical planes yield a branched map conjugate to a polynomial?...

Can polynomial intertwinings be constructed by quasiconformal surgery?...

For a fixed first polynomial $P_1$, does the polynomial obtained by intertwining $P_1$ with $P_2$ vary continuously with $P_2$?...

Is the Mandelbrot set locally connected? Equivalently, for the quadratic family $z\mapsto z^2+\lambda$, is the boundary of the unbounded component of ...

For a degree-$n$ Blaschke product $A$, let $B(A)$ be the rational maps obtained by mating $A$ with a varying Blaschke product, and let $F:B(z^n)\to B(...

Quotient the boundary of $B(A)$ by quasiconformal conjugacy, writing the quotient as $\partial(A)$. Prove that the natural isomorphism $F:B(z^n)\to B(...

Give a combinatorial description, possibly by laminations, of the quotient boundary space $\partial(z^n)$ obtained from the boundary of $B(z^n)$ by id...

Is $B(z^n)$ a domain of holomorphy? More generally, is every component of the space of expanding rational maps, or of expanding polynomials, a domain ...

For $f_\lambda(z)=\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove tha...

For $f_\lambda(z)=\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove tha...

Let $f_i(z)=z^2+c_i$ range over finitely many critically periodic quadratic polynomials, let $g_n$ be the iterated tuning $f_1\vdash\cdots\vdash f_n$,...

For $P_\rho'(z)=\lambda(1-z)^\rho$, $P_\rho(0)=0$, let $h$ linearize the Siegel disk and write $h'(\zeta)/(1-h(\zeta))=\sum_{\nu\ge0}a_\nu\zeta^\nu$. ...

For $P_\alpha(z)=z^2+e^{2\pi i\alpha}z$ with a Cremer fixed point at $0$, is there an arc in its Julia set joining $0$ to its preimage $-e^{2\pi i\alp...

Give a plausible topological model for the Julia set of a Cremer polynomial....

Produce a reliable computer picture of the Julia set of a Cremer polynomial....

If $J(f)=\widehat{\mathbb C}$, is $f$ ergodic for Lebesgue measure? At least, does it have at most $2\deg f-2$ ergodic components?...

Are Lattès maps the only rational maps having measurable invariant line fields on their Julia sets?...

In the setting of Przytycki Problem 1.1, is every $x\in\partial U$ with $\liminf_{n\to\infty}n^{-1}\log|(f^n)'(x)|>0$ accessible from $U$?...

For which positive-entropy invariant measures $m$ does failure of uniform $L^2(m)$ boundedness of the sums of $\log\operatorname{Jac}_m f-\kappa\log|f...

For which positive-entropy invariant measures $m$ does uniform $L^2(m)$ boundedness of the sums of $\log\operatorname{Jac}_m f-\kappa\log|f'|$, where ...

Which theorems about boundary behavior of Riemann maps have analogues for geometric coding trees?...

Find a collection of representative examples of transcendental entire maps whose dynamics may serve as models for general phenomena....

Describe the dynamics of Newton's method when applied to broad natural classes of transcendental entire functions....

Under the hypotheses of Eremenko–Lyubich Question 2, can even a single orbit converge to $z_0$?...

Call a polynomial bad if its Newton map has an attracting cycle that is not a root. Prove that every bad degree-$d$ polynomial $f_1$ belongs to a one-...

For $P_c(z)=z^2+c$ with connected Julia set, characterize the parameters $c$ for which $J(P_c)$ is locally connected....

Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for every periodic or preperiodic kneading sequ...

Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for arbitrary kneading sequences....

In the rank-zero case, if the limiting map on an invariant stable component is constant with value $x_0\in\partial U$, prove that one eigenvalue at $x...

Can the subsequential limit map on an invariant stable component of a polynomial diffeomorphism of $\mathbb C^2$ be a retraction onto a Herman ring or...

In the rank-two case for a polynomial diffeomorphism of $\mathbb C^2$, can an invariant stable component be a product of two Herman rings, or a produc...

For every degree $d$, prove that expanding (hyperbolic, Axiom A) maps are dense in the spaces $\operatorname{Rat}_d$ of rational maps and $\operatorna...

Develop an analogue of the Haken decomposition for geometrically finite rational maps. In particular, if the Julia set is disconnected, can the map be...

What is the combinatorial complexity of the Voronoi diagram of a set of lines (or line segments) in three dimensions?...

What is the complexity of the union of ``fat'' objects in $\mathbb{R}^3$?...

What is the maximum number of $k$-sets? (Equivalently, what is the maximum complexity of a $k$-level in an arrangement of hyperplanes?)...

Is there a deterministic, linear-time polygon triangulation algorithm significantly simpler than that of Chazelle?...

Is there an $O(n)$-space data structure that supports $O(\log n)$-time point-location queries in a three-dimensional subdivision of $n$ faces?...

Can the number of simple polygonalizations of a set of $n$ points in the plane be computed in polynomial time? A simple polygonalization is a simple p...

Given a visibility graph $G$ and a Hamiltonian circuit $C$, determine in polynomial time whether there is a simple polygon whose vertex visibility gra...

What is the complexity of the vertical decomposition of $n$ surfaces in $\mathbb{R}^d$, $d \ge 5$?...

Can a minimum-link path among polygonal obstacles be found in subquadratic time?...

Can an $n$-vertex polygonal curve be simplified in time nearly linear in $n$?...

How efficiently can one compute a polyhedral surface that is an $\epsilon$-approximation of a given triangulated surface in $\mathbb{R}^3$?...

Is the flip graph connected for general-position points in $\mathbb{R}^3$? Given a set of $n$ points in $\mathbb{R}^3$, the flip graph has a node for ...