Open-problem statements, with their sources attached.

Conjecture 1.3* (the $\times p, \times q$ conjecture): the only atomless Borel probability measure on $\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure.

The MLC conjecture, stating that the mandelbrot set is locally connected.

A stronger version of the MLC conjecture, stating that all multibrots are locally connected. Note that we don't need to require 2 ≤ n because the conjecture holds in the trivial cases n = 0 and n = 1 too.

The density of hyperbolicity conjecture, stating that the set of all parameters c for which fun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set.

The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all parameters c for which fun z ↦ z ^ n + c has an attracting cycle is dense in multibrotSet n. Note that we need to require 2 ≤ n because the conjecture is trivially false for n = 1.

The boundary of the Mandelbrot set is conjectured to have zero area.

The boundary of any Multibrot set is conjectured to have zero area. Note that we don't need to exclude the trivial cases n = 0 and n = 1 because the conjecture holds for them.

Gottschalk's surjunctivity conjecture* (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.

The Collatz conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is 1.

Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term. The Collatz conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is 1.

Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term. The Juggler Conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is $1$.