Weak Hyperbolicity on Periodic Orbits for Polynomials
Abstract
We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like $n^{5 + \epsilon}$, for some $\epsilon > 0$, 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2001
 arXiv:
 arXiv:math/0110155
 Bibcode:
 2001math.....10155R
 Keywords:

 Dynamical Systems
 EPrint:
 6 pages, Latex