2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62396We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like $n^{5 + ε}$, for some $ε> 0$, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with "small" multipliers. Somehow surprinsingly the proof is based in measure theorical considerations.6 pages, LatexDynamical Systemstext