Mathematics Problem Archive

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 $P_2$?...

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 ...

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....

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?...

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-...

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...

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

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 ...

Can every convex polytope in $\mathbb{R}^3$ be partitioned into tetrahedra such that the dual graph has a Hamiltonian path?...

Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? A light ray is trapped if it ...

How many intersections among an arrangement of pseudosegments in the plane must be added as vertices to allow the pseudosegment arrangment to be exten...

How many polyominoes on $n$ squares are there? A polyomino is a connected interior-disjoint union of axis-aligned unit squares joined edge-to-edge, in...

Is it true that every two sets of $n$ planar points in general position with the same number points on their convex hulls have compatible triangulatio...

For a planar point set $S$, is the number of pointed pseudotriangulations always at least the number of triangulations? A pseudotriangle is a planar p...

Consider a polyhedron with simply connected facets (no holes on a facet) and without boundary (every edge is incident to exactly two facets). Can the ...

Can every closed polyhedron be cut along its surface and unfolded into one piece in the plane without overlap? Such an unfolding is called a general u...

Does every simple graph with maximum vertex degree $\Delta \leq 6$ have a 3D orthogonal point-drawing with no more than two bends per edge? A 3D ortho...

What is the complexity of finding a tour of maximum Euclidean length for a planar point set?...

What is the complexity of finding a shortest tour in a solid planar grid graph? A planar grid graph is a graph whose vertices are any set of points on...

What is the complexity of the pallet loading problem? Given two pairs of numbers, $(A,B)$ and $(a,b)$, and a number $n$, decide whether $n$ small rect...

What is the optimal partition of a square into convex pieces such that the circularity of the pieces is optimized? The circularity of a polygon is the...