Mathematics Problem Archive

For real matrices $A\in\operatorname{Mat}_{n\times n}$, $B\in\operatorname{Mat}_{n\times p}$, and $C\in\operatorname{Mat}_{m\times n}$ with $n=mp$, de...

Does there exist a family of bounded-degree expander graphs, with spectral gap bounded away from zero and diameter tending to infinity, whose coarse R...

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...

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...

Are there quasiperiodic global minimals for metrics on the torus that are close to a flat three-dimensional torus? Here a geodesic is a global minimal...

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...

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...

Characterize mixing shifts of finite type up to topological orbit equivalence....

If a nonnegative integer matrix $A$ has a unique, simple, nonzero eigenvalue $n$, prove that $A$ is strong shift equivalent over $\mathbb Z_+$ to the ...

Classify shifts of finite type up to topological conjugacy; in particular, give a decision procedure determining whether two nonnegative integer matri...

Given a mixing shift of finite type $S_A$, determine the range of the dimension representation $\operatorname{Aut}(S_A)\to\operatorname{Aut}(G_A)$....

Let $S\subset\mathbb R$ be a unital subring and let $A$ be a square matrix over $S$ whose nonzero spectrum satisfies the spectral-conjecture condition...

Let $A,B$ be irreducible integer matrices of the same spectral radius. Suppose $\operatorname{tr}(A^n)>0$ implies $\operatorname{tr}(B^n)>0$ for every...

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-...

Prove that two mixing Markov shifts admit a magic-word isomorphism exactly when they have the same beta function, the same ratio group $\Delta$, and t...

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...

Suppose a $\mathbb Z^2$ action $\alpha$ has $\alpha^{\boldsymbol n}$ an SFT for some $\boldsymbol n$. Can $\alpha$ have infinitely many expansive comp...

If $S$ and $T$ are mixing shifts of finite type, prove that $S^i$ and $T^j$ can commute for all sufficiently large integers $i,j$....

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...

Determine all shift-invariant Borel probability measures and all subsystems of the symbolic system $X$ constructed in Section 14 of the source from th...

For $d>1$, if a continuous factor map sends a $\mathbb Z^d$ SFT $X$ onto a sofic shift $Y$, must $X$ contain a sofic subshift $W$ with $h(W)=h(Y)$ and...

Characterize the stable limit sets of one-dimensional cellular automata....

Let $T$ be a mixing sofic shift with a receptive fixed point. When is there a block code $f:T\to T$ and an SFT $T'\supset T$ such that $f(T')\subset T...

Let $f$ be a surjective block code from a mixing sofic shift $T$ to itself. When does there exist an SFT $T'\supset T$ such that $f(T')\subset T$?...

Does every nonabelian $\mathbb Z^d$ group shift factor algebraically onto a Bernoulli group shift?...

Is every nonabelian $\mathbb Z^d$ group shift weakly algebraically equivalent to a Bernoulli group shift?...

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

If a translation-invariant Markov random field $\mu$ on a shift is the unique Markov random field, even without assuming translation invariance, with ...

If a $\mathbb Z^d$ SFT has a unique measure of maximal entropy and that measure is Bernoulli, must an i.i.d. process map finitarily onto it?...

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...

Let $S$ be a one-sided subshift and $T$ the full one-sided shift on $N$ symbols, with $h(S)<\log N$, and suppose no point of $S$ has more than $N$ pre...

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

Give a procedure deciding whether two canonical left-resolving irreducible covers of a one-sided irreducible SFT are related by an automorphism carryi...

Are the groups $\operatorname{Aut}(\sigma_2)$ and $\operatorname{Aut}(\sigma_3)$ isomorphic?...

For a mixing SFT $S$, let $\operatorname{Aut}_0(S)$ be the inert automorphisms and $F_0(S)$ its finite-order-generated subgroup. Prove that $\operator...

Is every minimal action of a countable amenable group on the Cantor set topologically orbit equivalent to a $\mathbb Z$ action?...

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?...

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

For a surjective one-dimensional cellular automaton $f$ on the full $N$-shift, let $\nu(f,S_N)$ be the limsup exponential growth rate of points that a...

With $\nu(f,S_N)$ defined as in Question 25.3, must every surjective one-dimensional cellular automaton satisfy $\nu(f,S_N)\ge\sqrt N$?...

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...

If $\beta$ is a Salem number, prove that the periodic points of the beta-transformation $x\mapsto\beta x\pmod1$ are exactly $\mathbb Q\cap[0,1)$....