Mathematics Problem Archive

Let $\gamma$ be a smooth convex plane billiard table symmetric about an axis $l$, and let $C$ be a convex caustic. Must $C$ be symmetric about $l$? Pr...

A planar projective billiard is a bounded domain $\Omega$ whose piecewise-smooth boundary carries a transverse line field $L$. At $p\in\partial\Omega$...

Suppose a bounded strictly convex planar billiard has two nested closed caustics such that the smaller caustic is itself a caustic for the billiard in...

Let $\gamma\subset\mathbb{R}^n$ be a closed strictly convex hypersurface, and let $\Pi$ be the phase cylinder of oriented lines meeting $\gamma$ trans...

Let $C\subset\mathbb{R}^2$ be a curve and translate it through an $\varepsilon$-square lattice, recording a click whenever it meets a lattice point. C...

Let $g^t:\mathbb{R}^2\to\mathbb{R}^2$ be a Lebesgue-measure-preserving flow or cascade. A point is trapped if its positive semiorbit is bounded and it...

A uniformly massive planar body $B$ moves through a uniform medium of initially stationary infinitesimal particles, which reflect elastically from $\p...

A body moves freely in a rarefied medium in $\mathbb{R}^n$, $n\geq1$, under Newtonian aerodynamics. Determine the equations of motion and prove existe...

In $\mathbb{R}^n$, the space $\mathcal{L}$ of oriented lines has dimension $2n-2$ and a natural symplectic structure. Normal families of rays form Lag...

For a planar oval $\gamma$, alternately follow chords in two fixed directions to obtain a circle map $F:\gamma\to\gamma$. If $F$ is conjugate to a rot...

In a planar symplectic billiard on an oval, the chord $xy$ reflects to $yz$ when the tangent at $y$ is parallel to $xz$; define polygonal symplectic b...

For an oval $\gamma$ and a light source inside it, call the envelope of rays after $n$ reflections the $n$th caustic by reflection. Is every generic c...

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...

Given two monic polynomials of the same degree with connected filled Julia sets, form their topological mating by identifying their circles at infinit...

Can polynomial matings, including cases with infinite critical orbits, be constructed directly by quasiconformal cut-and-paste surgery?...

When one or both input polynomials in a mating vary continuously, does the resulting rational function vary continuously?...

For polynomials $P_1,P_2$ satisfying the tuning construction's connectedness and critical-basin hypotheses, is the resulting topological branched map ...

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 $P_2$?...

Let $B(z^n)$ be the set of degree-$n$ polynomials with an attracting fixed point whose immediate basin contains every critical point. Describe the bou...

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 ...

Starting with an orientation-preserving branched covering $f_0:S^2\to S^2$ and three marked base points, iteratively conjugate it as in Thurston's pul...

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 a quadratic Siegel polynomial with rotation angle $\theta$, find the arithmetic condition on $\theta$ that makes the Siegel-disk boundary a Jordan...

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

Analyze Julia and Fatou geometry at a general Misiurewicz parameter whose critical point never returns close to itself. To what extent does the real-q...

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....

Can any external ray land at a Cremer periodic point?...

Can the critical point of a Cremer polynomial be accessible from the complement of its Julia set?...

For a quadratic Cremer polynomial $P_\alpha$, how many connected components does $J(P_\alpha)\setminus\{0\}$ have? In particular, is the number counta...

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...

Can a Siegel disk have a boundary that is not a Jordan curve?...

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

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...

If a rational map $f$ has $J(f)=\widehat{\mathbb C}$, is $\omega(z)=\widehat{\mathbb C}$ for almost every $z$, and is $f$ conservative with respect to...

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?...

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 ...