Mathematics Problem Archive

Prove that the outer billiard about almost every convex polygon has an unbounded orbit....

Let $M$ be a compact negatively curved Riemannian manifold. Do the probability measures $|\varphi_j|^2\,d\operatorname{vol}$ associated with every ort...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Let $\gamma$ be a smooth closed strictly convex plane curve, parametrized by arc length $s$, and let $d>0$. Form a sphere-like surface by gluing the t...

Let $\gamma$ be a smooth closed strictly convex plane curve. The outer billiard map $T$ sends a point $A$ near $\gamma$ to the point $T(A)$ for which ...

Let $\gamma$ be a smooth closed strictly convex curve and $\delta\in(0,\pi/2)$. Say that $\gamma$ has the $\delta$-Gutkin property if the curve of inc...

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

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 $g^t:\mathbb{R}^2\to\mathbb{R}^2$ 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...

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 ...

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...

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...

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

For the quadratic Siegel polynomial with rotation angle $\theta_0=(\sqrt5-1)/2$, prove that the Siegel-disk boundary has the experimentally observed o...

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...

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...

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...