Weak Hyperbolicity on Periodic Orbits for Polynomials

الأصل · EN

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n⁵ ⁺ 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.