Boston–Jones dense-set conjecture for quadratic iterated Galois groups

Prove that for every irreducible quadratic polynomial over a number field, the arithmetic iterated Galois group is densely settled, meaning that it contains a dense set of elements whose proportion of vertices in stable cycles tends to one along the levels of the preimage tree.

Background

The paper recalls a conjecture of Boston and Jones concerning the asymptotic behavior of elements in arithmetic iterated Galois groups associated with irreducible quadratic polynomials over number fields. The conjecture asserts the existence of a dense set of elements exhibiting stable-cycle behavior on almost all vertices at sufficiently deep levels.

The paper notes that the conjecture remains unresolved in full generality, although it is known when the critical point is periodic or preperiodic with orbit length two. This makes the complete quadratic case an explicitly stated unresolved problem.

References

In , Boston and Jones conjectured that for every irreducible quadratic polynomial with coefficients in a number field $K$ the group $G_\infty(K,f,t)$ is densely settled, this is, there is a dense set of elements $g$ in $G_\infty(K,f,t)$ such that the proportion of vertices in level $n$ that are in stable cycles of $g$ approaches to $1$ when $n \rightarrow +\infty$. Although the conjecture has not been confirmed for every irreducible quadratic polynomial, it is known when the critical point of $f$ in $K$ is periodic or when it is preperiodic with orbit length $2$ .

— The inverse Galois problem of iterated Galois groups and their fixed-point proportion  (2608.14524 - Radi, 14 Aug 2026) in Section 1, Introduction