Weak hyperbolicity on periodic orbits for polynomials
Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1113-1118.

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n5+ε with the period, 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. Somewhat surprisingly the proof is based on measure theorical considerations.

On démontre que si les multiplicateurs des orbites périodiques répulsives d'un polynôme complexe croissent au moins comme n5+ε avec la période, où ε>0, alors l'ensemble de Julia du polynôme est localement connexe quand il est connexe. Comme conséquence on obtient que pour un polynôme complexe l'existence d'un cycle de Cremer implique l'existence d'une suite de cycles répulsifs ayant des multiplicateurs « petits ». D'une façon un peu surprenante la démonstration utilise des arguments de la théorie de la mesure.

Published online:
DOI: 10.1016/S1631-073X(02)02413-5
Rivera-Letelier, J. 1

1 Institute for Mathematical Sciences SUNY, Stony Brook, NY 11794-3651, USA
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Rivera-Letelier, J. Weak hyperbolicity on periodic orbits for polynomials. Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1113-1118. doi : 10.1016/S1631-073X(02)02413-5. http://www.numdam.org/articles/10.1016/S1631-073X(02)02413-5/

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