Mathematics Problem Archive

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...

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 ...

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...