Mathematics Problem Archive

Does every Cremer polynomial have Julia set of Hausdorff dimension two? Does every Cremer Julia set have Lebesgue measure zero?...

For a Cremer point of an arbitrary rational map, does every neighborhood contain infinitely many periodic orbits?...

Give an example of a Siegel polynomial whose Julia set is provably locally connected. Is the Julia set locally connected for Lebesgue-almost every Sie...

Does any rational function have a Siegel disk with a periodic point on its boundary?...

For every real $c\in[-2,1/4]$, is the Julia set of $f_c(z)=z^2+c$ locally connected?...

Does every nested intersection $\bigcap_k H_1*\cdots*H_k*M$ of tuned copies of the Mandelbrot set consist of one point? Equivalently, are infinitely r...

For the Mandelbrot limb $M(p/q)$ of internal angle $p/q$, is $\operatorname{diam}M(p/q)<K/q^2$ for an absolute constant $K$? If not, is it at least bo...

Find an explicit rational map whose Julia set has Hausdorff dimension two. When such a Julia set has zero Lebesgue measure, identify a natural geometr...

For an analytic family $\mathcal A$ of rational maps, let $Q\subset\mathcal A$ be the $J$-unstable locus. What is the Lebesgue measure of $Q$, and is ...

For a geometric coding tree of inverse branches of a holomorphic map, let $z_\infty:D(z_\infty)\to\overline U$ map each convergent symbolic branch to ...

Let $f:U\to f(U)$ be a proper holomorphic map of degree at least two on a simply connected attracting basin $U$, and suppose $f$ extends holomorphical...

In the setting of Przytycki Problem 1.1, is $h_{\mathrm{top}}(f|_{\partial U})=\log\deg(f|_U)$?...

Can the boundary of a Siegel disk contain periodic points or points with positive Lyapunov exponent? Must the topological entropy of the boundary dyna...

For a holomorphic quasi-repeller $\Lambda$, is every invariant ergodic measure on $\overline\Lambda$ the image of a measure on a one-sided shift under...

Characterize the positive-entropy invariant measures $m$ on a holomorphic quasi-repeller for which the almost-sure invariance principle, law of the it...

For a positive-entropy invariant measure $m$ on a holomorphic quasi-repeller $\Lambda$, is $m$ absolutely continuous with respect to Hausdorff measure...

For a holomorphic quasi-repeller $\Lambda$, is $\sup_{m\in\mathcal M^+(\Lambda)}\dim_Hm=\dim_H\overline\Lambda$? Does allowing all invariant ergodic m...

Describe the dynamics of the entire maps $z\mapsto\lambda e^z\sin z$ and $z\mapsto\lambda e^z\cos z$....

For $E_\lambda(z)=\lambda e^z$, characterize completely the parameters $\lambda$ for which $J(E_\lambda)=\mathbb C$....

Many parameters with $J(E_\lambda)=\mathbb C$ lie on parameter curves or hairs. Are these hairs $C^\infty$? Are they analytic?...

For parameters $\lambda,\mu>1/e$, are the Knaster-like continua arising in the dynamics of $E_\lambda(z)=\lambda e^z$ and $E_\mu(z)=\mu e^z$ homeomorp...

Describe the parameter-space structure for the entire families $C_\lambda(z)=\lambda\cos z$ and $S_\lambda(z)=\lambda\sin z$....

Determine the measure and Hausdorff dimension of the parameter hairs in the exponential, sine, and cosine families....

Does there exist an entire function with a wandering Fatou component whose orbit of components is bounded?...

Let $\varphi$ be a holomorphic germ fixing $z_0$ with multiplier $e^{2\pi i\alpha}$ for irrational $\alpha$. Can $\varphi^n(z)\to z_0$ uniformly on so...

Let all roots of a degree-$d$ polynomial $f$ lie in the unit disk, let $\alpha$ be a root of multiplicity $m$, and let $A^*_{h}(\alpha)$ be its immedi...

If the quadratic polynomial $P_c(z)=z^2+c$ is nonrecurrent, does there exist a conformal metric $\rho(z)|dz|$ with integrable singularities in which $...

For monic polynomials $z^n+a_{n-1}z^{n-1}+\cdots+a_1z$ with $|a_1|\ge1$, external rays landing at the fixed point $0$ have a rotation number. Does thi...

For a piecewise monotone interval map, iteratively replace its critical values by those of a polynomial with the same ordered critical data and conjug...

For the lift family $x\mapsto k-k|2x-1|^\alpha$, $\alpha>1$, does the real Thurston algorithm converge whenever the initial interval map has a periodi...

Let $f$ be a polynomial diffeomorphism of $\mathbb C^2$ with Jacobian determinant $\delta$, let $U$ be a component of the interior of the bounded-forw...

For a geometrically finite rational map $f$, prove that either its Julia set is the whole sphere and $f$ is ergodic there, or its Julia set has Hausdo...

Prove that every connected component of the Julia set of a geometrically finite rational map is locally connected....

Extend Thurston's finite combinatorial classification from critically finite rational maps to all geometrically finite rational maps: give finite topo...

If a complete hyperbolic $3$-manifold $N$ has fundamental group generated by $n$ elements, is there a bound $R_n$, depending only on $n$, on the radiu...

For $n>1$, do there exist nontrivial critically finite rational maps $f:\mathbb P^n\to\mathbb P^n$ whose postcritical hypersurface $V$ has Kobayashi-h...

Let $A$ be a hyperbolic attractor of a diffeomorphism of a compact $3$-manifold. Beyond the known Anosov, laminated, Williams, and invariant-torus cas...

Given an explicitly specified smooth orientation-preserving diffeomorphism $F$ of the $2$-sphere, is its topological entropy Turing-computable to arbi...

Can the Euclidean minimum spanning tree (MST) of $n$ points in $\mathbb{R}^d$ be computed in time close to the lower bound of $\Omega(n \log n)$?...

What is the complexity of computing a minimum-cost Euclidean matching for $2n$ points in the plane? The cost of a matching is the total length of the ...

Can the class of 3SUM hard problems be solved in subquadratic time? These problems can be reduced from the problem of determining whether, given three...

What is the best output-sensitive convex hull algorithm for $n$ points in $\mathbb{R}^d$?...

How many $\pi$-floodlights are always sufficient to illuminate any polygon of $n$ vertices, with at most one floodlight placed at each vertex? An $\al...

Can the interior of every simply connected polyhedron whose surface is meshed by an even number of quadrilaterals be partitioned into a hexahedral mes...

For a point set $P$ in $\mathbb{R}^d$, let $f_d(P)$ be the number of unit-distance point pairs: $$f_d(P) = \left| \{ (u,v) \mid u, v \in P, \, \|u-v\|...

For any (planar) triangle $T$, is there is a $3$-coloring of the (infinite) plane with no monochromatic copy of $T$? We imagine congruent copies of $T...

Is there a data structure maintaining a set of $n$ points in the plane subject to insertions, deletions, and nearest-neighbor queries in $O(\log n)$ t...

Let $\rho(S)$ be the fewest number of reflex vertices in a polygonization of a 2D point set $S$, i.e., the fewest reflexivities of any simple polygon ...

Is the Yao-Yao Graph a $t$-spanner for constant $t$? A geometric graph is a $t$-spanner (or just a spanner) if, for every pair of nodes, the shortest ...

Identify or construct all $k$-equiprojective polyhedra. A polyhedron $P$ is $k$-equiprojective if its orthogonal projection to a plane is a $k$-gon in...