A Complex Gap Lemma

A complex Gap lemma

Sébastien Biebler (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets of the plane associated to a holomorphic IFS. Our main result is a complex version of Newhouse's Gap Lemma : we show that under some assumptions, if the product t(K)t(L) of the thicknesses of two Cantor sets K and L is larger than 1, then K and L have non empty intersection. Since in addition this thickness varies continuously, this gives a criterion to get a robust intersection between two Cantor sets in the plane.

Domaines: Systèmes dynamiques [math.DS]

Sébastien Biebler. A Complex Gap Lemma. Proceedings of the American Mathematical Society, 2019, 148 (1), pp.351-364. ⟨10.1090/proc/14716⟩. ⟨hal-01887360⟩