Mathematics Problem Archive

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

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...

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...

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

Is every strongly mixing measure-preserving transformation strongly mixing of order three?...

Starting from a positive integer $a_0$, define $a_{n+1}=\lfloor a_n^{1/2}\rfloor$ when $a_n$ is even and $a_{n+1}=\lfloor a_n^{3/2}\rfloor$ when $a_n$...

Is the Mandelbrot set locally connected? Equivalently, for the quadratic family $z\mapsto z^2+\lambda$, is the boundary of the unbounded component of ...

For $f_\lambda(z)=\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove tha...

For $f_\lambda(z)=\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove tha...

For the quadratic Siegel polynomial with rotation angle $\theta_0=(\sqrt5-1)/2$, prove that the Siegel-disk boundary has the experimentally observed o...

Produce a reliable computer picture of the Julia set of a Cremer polynomial....

If a quadratic polynomial $f_c$ is infinitely renormalizable of bounded type, must $J(f_c)$ be locally connected? In particular, is the Julia set of t...

For every real $c\in[-2,1/4]$, is the Julia set of $f_c(z)=z^2+c$ locally connected?...

Does every nested intersection $\bigcap_k H_1*\cdots*H_k*M$ of tuned copies of the Mandelbrot set consist of one point? Equivalently, are infinitely r...

Can a nowhere-dense Julia set of a rational map have positive Lebesgue measure?...

Are Lattès maps the only rational maps having measurable invariant line fields on their Julia sets?...