Mathematics Problem Archive

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 $...

Consider $$x_{n+k}=\frac{A_0+A_1x_n+\cdots+A_kx_{n+k-1}}{B_0+B_1x_n+\cdots+B_kx_{n+k-1}},$$ where the coefficients are nonnegative, $\sum A_i,\sum B_i...

Consider a Hamiltonian system with a center at the origin and Hamiltonian $$H(x,y)=H_{2n}(x,y)+H_m(x,y),\qquad m>2n,$$ where $H_{2n}$ and $H_m$ are ho...

For $$\dot x=P_{2k+1}(x,y),\qquad \dot y=Q_{2\ell+1}(x,y),$$ where $P_{2k+1}$ and $Q_{2\ell+1}$ are homogeneous polynomials of the indicated odd degre...

Let $\mathcal T(n)$ be the maximum number of critical periods that a planar polynomial differential system of degree $n$ can have. Is there a constant...

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be smooth, have a fixed point, and satisfy $$\rho\bigl(|DF(x)|\bigr)<1\quad\text{for every }x\in\mathbb{R}^2.$$ Is...

Let $A_0,\ldots,A_n$ be independent $N(0,1)$ random variables and let $p_n$ be the probability that the zero solution of $$A_nx^{(n)}+A_{n-1}x^{(n-1)}...

Do there exist two irreducible algebraic curves of degrees $n$ and $m$, with $n+m>4$, each having an oval, for which the Poncelet map is well defined ...

Let $\gamma=\{x^2+y^2-1=0\}$ and $\Gamma_\varepsilon=\{p_2(x,y)+\varepsilon p_m(x,y)=0\}$, where $\Gamma_0$ is an ellipse surrounding $\gamma$, the cu...

Let $f(x_1,\ldots,x_n)\in\mathbb{C}[x_1,\ldots,x_n]$ satisfy $$M_m:=\int_0^1\cdots\int_0^1 f(x_1,\ldots,x_n)^m\,dx_1\cdots dx_n=0\qquad(m\geq1).$$ Mus...

Let $f(x)\in\mathbb{C}[x]$ have $k$ monomials. Does there exist $N(k)$ such that if $$M_n:=\int_0^1f(x)^n\,dx=0\qquad(1\leq n\leq N(k)),$$ then $f=0$?...

Define $f:\mathbb{N}\to\mathbb{N}$ by $f(n)=n+\operatorname{rev}(n)$, where $\operatorname{rev}$ reverses the decimal digits. Are there infinitely man...

Determine the structure of the multiple correlation sequences $(\mathcal{C}(n_1,\ldots,n_\ell))$ defined by (source reference E:MultCor). Is it true t...

Is there a sequence that is good for $2$-recurrence of powers but is not good for $2$-recurrence of commuting transformations?...

Give an explicit example of a fast growing sequence that is good for multiple recurrence and convergence of powers and commuting transformations....

Let $\mathcal P$ be an essentially distinct family of integer polynomials, and let $d_{\min}(\mathcal P)$ be the least $d$ for which the Host–Kra fact...

Let $(X,\mathcal X,\mu,T)$ be a system and $f,g\in L^\infty(\mu)$ be functions. If $\Lambda$ is the von Mangoldt function and $\phi$ is a multiplicati...

Let $p_1,\ldots, p_\ell$ be integer valued generalized polynomials. Show that the averages $$ \frac{1}{N}\sum_{n=1}^{N} T_1^{p_1(n)}f_1\cdots T_\ell^{...

Let $(X,\mathcal X,\mu,T_1,\ldots, T_\ell)$ be a system and $\{p_1,\ldots, p_\ell\}$ be a family of intersective integer polynomials. Show that for ev...

Let $(X,\mathcal X,\mu)$ be a probability space, $T_1,\ldots, T_\ell \colon X\to X$ be invertible measure preserving transformations, and $p_1,\ldots,...

Find an example of a function $a\in \mathcal H$ that grows faster than polynomials, meaning, $a(t)/t^k \to\infty$ for every $k\in \mathbb N$, such tha...

Suppose that $n\sigma_n\to\infty$. Show that almost surely the sequence $(a_n(\omega))$ is good for multiple recurrence and convergence of commuting t...

Suppose that $a,b\in(0,1)$ and $a\neq b$. Show that almost surely the following holds: For every system $(X,\mathcal X,\mu,T,S)$ and functions $f, g \...

Let $(X,\mathcal X,\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\in \mathcal X$ with $\mu(A)>0$. Is it true that ther...

Let $(X,\mathcal X,\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\in \mathcal X$ with $\mu(A)>0$. Is it true that ther...

For a smooth dissipative dynamical system with a global attractor, is the supremum of the local Lyapunov dimension on the attractor attained at an equ...