Mathematics Problem Archive

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 $S$ is an expansive automorphism of an irreducible shift of finite type, must $S$ itself be a shift of finite type?...

Prove that every expansive automorphism of a one-sided full shift is topologically conjugate to a two-sided full shift....

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 $d>1$, must every $\mathbb Z^d$ sofic shift be a factor of a $\mathbb Z^d$ shift of finite type having the same entropy?...

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?...

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...

Prove that the jointly periodic points of every surjective one-dimensional cellular automaton are dense....

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 $\nu(f,S_N)\ge\sqrt N$?...

Prove that for some $N>1$ there is a surjective one-dimensional cellular automaton $f$ with $\nu(f,S_N)<N$....

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 $\mathbb Q\cap[0,1)$....

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

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

Which pairs of abelian groups occur as $K_0(M,I)$ and $K_1(M,I)$ for a canonical matrix system of a subshift?...

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...

Consider the planar cubic rigid systems $$\begin{cases}\dot x=-y+x(a+bx+cy+dx^2+exy),\\ \dot y=x+y(a+bx+cy+dx^2+exy).\end{cases}$$ Is $2$ the maximum ...

Is $(n+m)/2$ the maximum number of limit cycles of $$\dot x=P_n(x,y),\qquad \dot y=Q_m(x,y),$$ where $n\neq m$ and $P_n,Q_m$ are homogeneous polynomia...

(i) For the cubic family $$\begin{cases}\dot x=ax+by,\\ \dot y=cx^3+dx^2y+exy^2+fy^3,\end{cases}$$ is $2$ the maximum number of limit cycles? (ii) If ...

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...

For $$\frac{dx}{dt}=(a_0+a_1\sin t+a_2\cos t)x^3+(b_0+b_1\sin t+b_2\cos t)x^2,$$ is $3$ the maximum number of $2\pi$-periodic limit cycles?...

Given integers $p>q\geq2$ and $m,n\in\mathbb{N}$, find the maximum number of $2\pi$-periodic limit cycles of $$\frac{dx}{dt}=A_m(t)x^p+B_n(t)x^q,$$ wh...