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Weak Hyperbolicity on Periodic Orbits for Polynomials

Juan E. Rivera-Letelier
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Abstract

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.

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Juan E. Rivera-Letelier. 2001-10-15. Weak Hyperbolicity on Periodic Orbits for Polynomials. https://arxiv.org/abs/math/0110155

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