12 problems

Let $f_c(x)=x^2+c$ be the unique real quadratic map whose critical orbit $0=x_0\mapsto x_1\mapsto\cdots$ has closest recurrence at the Fibonacci values, meaning … and suppose the c…

Let the quadratic family be the parameterized family of quadratic maps, and call an invariant probability measure an acip when it is absolutely continuous with respect to Lebesgue…

Let $q$ be an odd prime power and let $n\geq 3$. A quadratic system is a map … where each $f_i\in\mathbb{F}_q[x_1,\dots,x_n]$ has degree at most $2$. Classification conjecture. Suc…

Let $A(m,n)$ and $B(m,n)$ be the integer forms parametrizing rational $3$-cycles of maps $f_c(x)=x^2+c$, and let $(m,n)\in\mathbb{Z}^2$ satisfy $mn(m+n)\neq0$ and $\gcd(m,n)=1$. Nu…

Consider the quadratic map … with $t=0,1,2,\ldots$, and its periodic and preperiodic points. Preperiodic-point extraction conjecture. Exploration of preperiodic points should provi…

Consider the classification of figures by dynamic type and topological shape, and let a totally non-hyperbolic unimodal case mean a unimodal combinatorial case that is totally non-…

Let ${\bf h}(f)$ denote the topological entropy of a quadratic map $f$, and let a constant-entropy locus be a set of parameters in the unimodal region on which ${\bf h}$ is constan…

Let a minimal unimodal combinatorics be a minimal combinatorial type in the unimodal region, and call it unobstructed when it is realized by the corresponding quadratic dynamical s…

Let $q_c(z)=z^2+c$ be a quadratic map, let $\mathbb{M}$ denote the Mandelbrot set, and let a parameter $c$ be hyperbolic when the orbit of the origin accumulates on an attracting c…

Benedetto et al.'s 14-point conjecture. One has

No-long-cycles conjecture. The map $phi$ has no $\mathbb{Q}$-periodic point of period greater than or equal to $3$.

For each $d\geq 1$, let $M(d)$ denote the minimal uniform bound in the statement that every quadratic polynomial $\varphi_c(z)=z^2+c$ over a number field of degree at most $d$ has…