Mathematics Problem Archive

Are there geodesic flows or Birkhoff billiards with fractal caustics? More specifically, for every $1\leq s<2$, is there a caustic of a convex billiar...

For a Hamiltonian system, call the good set the maximal invariant subset on which the invariant Liouville measure is almost periodic; call the bad set...

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...

Is there an analogue of Pesin theory for suitably defined smooth maps of the objects that arise naturally in algebraic dynamical systems—compact sets ...

Let $X=\{0,1\}^{\mathbb{Z}}$ and $Y=\{y\in\{0,1,2\}^{\mathbb{Z}}:y_i\neq y_{i+1}\}$. Both are mixing shifts of finite type with entropy $\log2$, but t...

Let $T:[0,1)\to[0,1)$ be the doubling map $x\mapsto2x\pmod1$, and let $\mu$ be an ergodic measure for $T$ with $0<h(\mu)<1$. Call $f:\mathbb{R}\to\mat...

Let $A=\{0,1,\ldots,d-1\}$ and let $\sigma$ be the shift on $A^{\mathbb{Z}}$. Define $(v_n(x))_i=\#\{0\leq j\leq n:x_j=i\}$ and $(w_n(x))_i=\#\{0\leq ...

With $\mathcal{F}^+$ and $\mathcal{F}^-$ defined from the forward and backward symbol-count tail fields on the full shift $A^{\mathbb{Z}}$, if $\mathc...

Let $(M,T)$ be a smooth map on a manifold with a good symbolic cover: a mixing shift of finite type factors onto $(M,T)$ and is injective on a set of ...

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

Let $S$ and $T$ be subshifts. If $S$ is a mixing shift of finite type and $T$ is topologically orbit equivalent to $S$, must $T$ also be a mixing shif...

Consider billiard tables formed by two concentric semicircles joined by two line segments. Consider trajectories with a fixed circle, concentric with ...

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

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 stochastic zeta functions of mixing Markov shifts....

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?...

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

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?...

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?...

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?...

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 $\nu(f,S_N)<N$....

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

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

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 $$\frac{dx}{dt}=(a_0+a_1\sin t+a_2\cos t)x^3+(b_0+b_1\sin t+b_2\cos t)x^2,$$ 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...

Let $\mathcal M_m$ be the family of planar polynomial vector fields that are linear combinations of $m$ distinct monomial vector fields $(x^{n_j}y^{k_...

Let $f$ be a continuous, nonzero, $T$-periodic function and let $p>0$. Find necessary and sufficient conditions on $f$ for the existence of positive $...