Mathematics Problem Archive

Let the metric of a Markov space evolve by $$\frac{d}{dt}d(x,y)=-\kappa(x,y)d(x,y),$$ where $\kappa$ is computed from the current metric, with either ...

Replace the $T_1$ distance in the coarse-Ricci definition by $L^\infty$ transport, requiring a coupling that moves every point by at most $d(x,y)$. Do...

Do spaces with positive sectional curvature in the sense of Alexandrov have positive coarse Ricci curvature for a natural choice of Markov kernels? Ca...

What is the measure of the set of initial conditions of the Newtonian $N$-body problem that lead to global solutions? The complementary set of singula...

If two Birkhoff billiard maps $T_1$ and $T_2$ satisfy $T_1=ST_2S^{-1}$ for a homeomorphism $S$, must their tables be similar? Relatedly, can one hear ...

(a) Is the set of $n$-periodic orbits of a smooth strictly convex Birkhoff billiard nowhere dense for every $n$? (b) Does every polygonal Birkhoff bil...

Does a free gas coupled to a convex rigid container by conservation of momentum converge weakly to equilibrium, with the container—which moves only by...

In the space of area-preserving $C^1$ diffeomorphisms of a compact manifold, is it generic that the dynamics is either hyperbolic or has zero Lyapunov...

For the infinite-dimensional Calogero–Moser system, in which particles on the real line interact through the inverse-square potential, does the dynami...

Let $p$ be a prime for which $$f(u_1,u_2)=1+u_1u_2+u_1^2u_2+u_1^3u_2+u_1^4+u_2^2+u_1^4u_2^2$$ is irreducible, and consider the algebraic $\mathbb{Z}^2...

For $\mathbb{Z}^d$-actions by automorphisms of a connected group, mixing actions are mixing of all orders. Can this result be proved using simpler ide...

Choose a random subset $Q$ of the primes by independently retaining each prime with probability $1/2$. Is it almost surely true that $$\limsup_{n\to\i...

Given $\varepsilon>0$, does there exist a polynomial $f(x)=\prod_{i=1}^d(x-\alpha_i)\in\mathbb{Z}[x]$ whose logarithmic Mahler measure $$m(f)=\sum_{i:...

Let $\log_p:\mathbb{C}_p^*\to\mathbb{C}_p$ be the branch of the $p$-adic logarithm with $\log_p(p)=0$, and let $T_\lambda:x\mapsto\lambda x$ on $\math...

Let $M$ be a compact Riemannian manifold of constant negative curvature and $(g_t)_{t\in\mathbb{R}}$ its geodesic flow. Does there exist a probability...

For sofic shifts $S,T$ with $h(S)\ge h(T)$, give necessary and sufficient conditions for a factor map from $S$ onto $T$. The most fundamental unequal-...

Characterize the functions that occur as beta functions of mixing Markov shifts....

For a $\mathbb Z^2$ shift of finite type $\alpha$, characterize the possible sets $E_1(\alpha)$ of expansive directions, especially under the assumpti...

For multiplicatively independent integers $p,q>1$, can there exist a nonatomic Borel probability measure on the circle, other than Haar measure, that ...

For the symbolic system $X$ constructed in Section 14 of the source from a commuting pair of full-shift endomorphisms, is Haar measure its only invari...

Classify group shifts up to topological conjugacy, especially abelian group shifts with completely positive entropy....

Determine whether algorithms exist to: decide conjugacy of two SFTs; decide conjugacy of two two-sided or one-sided sofic shifts; compute the expansiv...

Let $T$ be a mixing one-sided SFT and $S$ a subshift with $h(S)<h(T)$. Give useful necessary and sufficient conditions for $S$ to be isomorphic to a s...

Classify irreducible SFTs up to topological orbit equivalence; characterize and classify their unital ordered cohomology groups; determine when windin...

Can a $C^r$ diffeomorphism of a compact Riemannian manifold have infinite symbolic-extension entropy?...

For each $1\le r\le\infty$, characterize the entropy structures that can occur for $C^r$ diffeomorphisms of compact Riemannian manifolds....

Must every surjective $d$-dimensional cellular automaton have dense periodic points? Must its jointly spatially and temporally periodic points be dens...

For a $d$-dimensional cellular automaton with $d\ge2$, let $f_P$ and $f_F$ be its restrictions to spatially periodic and finite configurations. Does i...

Characterize the countable groups $G$ for which every injective $G$-equivariant cellular automaton on $A^G$, for every finite alphabet $A$, is surject...

Is every irreducible sofic shift topologically conjugate to a renewal shift?...

Classify two-sided sofic shifts up to topological conjugacy....

Classify subshifts up to flow equivalence....

Prove that the substitutive shift of every irreducible Pisot substitution has pure discrete spectrum....

Let $x\in A^{\mathbb Z^2}$ and let $N_x(n_1,n_2)$ be the number of distinct $n_1\times n_2$ rectangular patterns in $x$. If $N_x(n_1,n_2)\le n_1n_2$ f...

For the classical polynomial Liénard system $$\dot x=y-F(x),\qquad \dot y=-x,$$ let $\operatorname{Lie}(n)$ be the maximum number of limit cycles when...

For a general $T$-periodic Riccati equation $$\frac{dx}{dt}=A_2(t)x^2+A_1(t)x+A_0(t),$$ give effective criteria determining whether it has a continuum...

Determine the maximum number $\mathcal{C}_n$ of centers for planar polynomial differential systems of degree $n\geq4$....

For the family of reversible quadratic centers $$\begin{cases}\dot x=-y+xy,\\ \dot y=x+Dx^2+Fy^2,\end{cases}$$ is $2$ the maximum number of critical p...

Is the period function associated with the period annulus of the origin for $$\dot z=iz+(z\bar z)^n z^{k+1},$$ where $n$ and $k$ are positive integers...

Determine the entries currently marked unknown in the following comparison between quadratic systems and planar piecewise-linear systems with a straig...

Let $\mathcal H(n)$ be the maximum number of limit cycles of degree-$n$ planar polynomial systems, and let $\mathcal L(n)$ be the maximum number of cr...

Are there smooth vector fields in $\mathbb{R}^3$ satisfying the hypotheses of the Markus–Yamabe conjecture and having periodic orbits?...

Does every triangular billiard have a periodic trajectory?...

Let $f$ be real analytic near the origin, with $f(0,0)=0$, and let $n>1$. Suppose the origin is an isolated equilibrium of $$\dot x=2^n\operatorname{R...

Find a reasonable, or sharp, upper bound in terms of $m_1,m_2$ for the maximum number of simple positive-coordinate solutions of a real polynomial sys...

Is $(2m_1-1)(2m_2-1)$ the maximum number of simple solutions of a real polynomial system $f_1(x,y)=f_2(x,y)=0$, where $m_i$ is the number of monomials...

For $n\in\mathbb{N}$, let $\Pi(n)$ be the product of its decimal digits, and let $\operatorname{Pm}(n)$ be the least positive integer such that $\Pi^{...

Let $\mathcal C_{T,S}$ be the set of sequences $(\int f\,T^ng\,S^nh\,d\mu)_{n\ge1}$ over probability-preserving systems with commuting $T,S$ and bound...

If $(a_1(n)),\ldots, (a_\ell(n))$ are sequences of integers, then show that the following %%three statements are equivalent: • The sequences $(a_1(n))...

Let $(a(n))$ be a sequence that satisfies: • for every connected $\ell$-step nilmanifold $X$ and every irrational nilrotation $b$ in $X$ the sequence ...