Weak Hyperbolicity on Periodic Orbits for Polynomials
Abstract
Description
We 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, Latex
6 pages, Latex