Mathematics Problem Archive

If a sequence is good for $2$-convergence of powers, then show that it is good for $2$-convergence of commuting transformations....

Suppose that the sequence of $\ell$-tuples of polynomials $(p_{1,N},\ldots, p_{\ell,N})$ is good. Show that for every ergodic system $(X,\mathcal X,\m...

Let $(X,\mathcal X,\mu,T)$ be a system and $f, g, h\in L^\infty(\mu)$ be functions. Show that the averages $$ \frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(T...

Let $(a(n))$ be the sequence of integers $(p_n)$, where $p_n$ is the $n$-th prime, or $([n^c])$ where $c>0$, or $(2^n)$. Is it true that for every erg...

Suppose that the polynomials $p_1,\ldots,p_\ell\in \mathbb Z[t]$ are rationally independent and have zero constant term. Show that for every $A\in \ma...

Let $(X,\mathcal X,\mu, T,S)$ be a system and $f,g\in L^\infty(\mu)$ be functions. Show that the averages $$ \frac1N \sum_{n=1}^N f(T^nx)\cdot g(S^nx)...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Let $a,b$ be distinct positive non-integers. Show that for every ergodic system $(X,\mathcal{X},\mu,T)$ and functions $f, g \in L^\infty(\mu)$, we hav...

Show that the sequence $([n \sin n])$ is good for multiple recurrence and convergence of powers....

Show that if $c>1$ is not an integer, then the sequence $([n^c])$ is good for multiple recurrence and convergence of commuting transformations. Moreov...

Let $\ell \in \mathbb N$ and $c,c_1,\ldots, c_\ell$ be positive real numbers. Show that the prime numbers contain patterns of the form $$ \{m,m+[n^{c}...

Suppose that $n\sigma_n\to\infty$. Show that almost surely the following holds: For every system $(X,\mathcal X,\mu, T,S)$ and functions $f, g \in L^\...

For a Hamiltonian diffeomorphism of a closed symplectic manifold, prove the Arnold lower bound on its number of fixed points in terms of Morse-theoret...

For a generic quantum system whose classical counterpart is integrable, prove that the unfolded high-energy level spacings have Poisson statistics....

Does there exist an ergodic measure-preserving transformation whose Koopman operator has simple Lebesgue spectrum?...

Under the hypotheses in which the Lyapunov (Kaplan–Yorke) dimension is defined from the ordered Lyapunov exponents, prove that it equals the appropria...

Classify invariant ergodic probability measures for higher-rank diagonalizable group actions on homogeneous spaces; in particular, prove that the meas...

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...