50 problems

Let $N\geq 3$, let $A_j(t)$, for $j=2,3,\ldots,N-1$, be polynomial functions, and consider the differential equation … A solution is a center at the origin if nearby solutions have…

Let $f\in\mathbb{Z}[x]$ be a quadratic polynomial. Call $f$ stable if every iterate $f^n$ is irreducible over $\mathbb{Q}$, and call $f$ critically infinite if the forward orbit of…

Let $f$ be a polynomial of degree $d$ with an indifferent fixed point at the origin. Let $c_i$ be the critical points of $f$, let $\theta$ be the rotation number at the origin, and…

Hyperbolic polynomial branch and extension conjecture. The following are equivalent:

Let $F=F_{a,v}$ be a cubic polynomial in the superattracting period-$p$ parameter curve $4\mathcal{S}_p4$, written in the normal form … Assume that $F$ is a Misiurewicz map, meanin…

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Dual Smale's mean value conjectu…

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Smale's mean value conjecture. T…

Let $A=x^n+x+1$ with $n\geq 4$, and let $s$ be the greatest positive integer such that … Consider the even and odd sequences associated with $A$ under the binary polynomial Collatz…

Let $A=x^n+x+1$ with $n\geq 4$, and let $s$ be the greatest positive integer such that … For each positive integer $t\leq s-1$, denote by $d_t$ the common degree asserted for the r…

Let $M_1=x^2+x+1$, and let the odd sequence of a polynomial be the sequence of odd polynomials obtained by the binary polynomial Collatz transformations. Let $r\geq 1$ and…

Let $A\in\mathbb{F}_2[x]\setminus\{0\}$. Define the associated sequences $(a_{2k})_k$, $(A_{2k})_k$, and $(A_{2k+1})_k$ by $A_0=A$, $A_{2k}=1+(x^2+x+1)A_{2k-1}$, and by removing al…

The colliding orbits conjecture. The set $C_f(\alpha_1,\alpha_2;\beta)$ is infinite if and only if $f(\alpha_1)=f(\alpha_2)$. This problem concerns when the strict forward orbits o…

Let $P$ be a polynomial having a parabolic cycle, and let $J(P)$ denote its Julia set. A small perturbation of $P$ should convert the immediate basin of the parabolic cycle into an…

Let $f:\mathbb{C}\to\mathbb{C}$ be a polynomial, and let a Fatou component mean a connected component of the Fatou set of $f$. For $\epsilon>0$, measure diameters in the usual Eucl…

Let $p$ be a polynomial of degree $d\geq 2$. Assume that $p(\alpha)=0$, and let $T_{p,1}$ be the corresponding Traub's map. Denote by $\mathcal A^{\star}(\alpha)$ the basin of attr…

Same-degree intersection-of-orbits conjecture. At least one of the following holds: (1) there exists $m\in{\mathbb N}$ such that $f^m=g^m$; or (2) there exist linear polynomials…

Intersection-of-orbits conjecture. At least one of the following holds: (1) there exist $m,n\in{\mathbb N}$ such that $f^m=g^n$; or (2) there exist linear polynomials…

Let $f_c(z)=z^2+c$. For positive integers $m,n$ with $m\mid n$, define the delta factors $\Delta_{n,m}$ by … and, when $m=n$, by … Here $\Phi^{\mathrm{cyc}}_k$ is the $k$-th cyclot…

Let an FDL be a finite dynamical lamination for degree $d$ polynomials, and let its free criticality be the number of critical polygons, counted with multiplicity, that fit outside…

Let $T\colon\mathbb{F}_2[x]\to\mathbb{F}_2[x]$ be the polynomial Collatz map, with $T(f)=(1+x)f+1$ for odd $f$ and $T(f)=f/x$ for even $f$. For polynomials of degree $d$, define th…

Let $\theta\in\mathbb{R}/\mathbb{Z}$. Let $\mathbf{P}^3$ be the space of conformal conjugacy classes of cubic polynomials, let $\mathbf{H}^3\subset\mathbf{P}^3$ be the hyperbolic c…

The fixed-point basis conjecture. For every $i\in\{1,2,\dots,k\}$, the polynomial

The dimensions conjecture. For every divisor $m$ of $\operatorname{lcm}(\deg R_1,\dots,\deg R_k)$, this vector space has dimension

Let $A=x^n+x+1$ with $n\geq 2$, and let $\ell_A$ denote the length of its binary-polynomial Collatz transformation sequence. Let $s$ be the greatest integer such that…

Let $A=x^n+x+1$ with $n\geq 2$, and let $A_1,A_3,\ldots$ be the odd subsequence in the binary-polynomial Collatz transformation. Let $s$ be the greatest integer such that…