Settled elements in profinite groups

Given a polynomial of degree d over number field, the image associated arboreal representation absolute Galois group field is profinite acting on d-ary tree. Boston and Jones conjectured that for quadratic polynomial, such contains dense set settled elements. Here an element if it exhibits certain pattern growth cycles at finite levels In this paper, we prove conjecture generically in case when has strictly pre-periodic post-critical orbit length 2, provide new evidence holds polynomials with orbits least 3. To our results, introduce dynamical method, which uses notions maximal torus its Weyl group. These are analogous to tori groups theory compact Lie groups, where they fundamental. For contain information about elements, foundation method.