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Wandering domains arising from Lavaurs maps with siegel disks

Matthieu Astorg, Luka Boc Thaler and Han Peters

Vol. 16 (2023), No. 1, 35–88
Abstract

The first example of polynomial maps with wandering domains was constructed in 2016 by the first and last authors, together with Buff, Dujardin and Raissy. In this paper, we construct a second example with different dynamics, using a Lavaurs map with a Siegel disk instead of an attracting fixed point. We prove a general necessary and sufficient condition for the existence of a trapping domain for nonautonomous compositions of maps converging parabolically towards a Siegel-type limit map. Constructing a skew-product satisfying this condition requires precise estimates on the convergence to the Lavaurs map, which we obtain by a new approach. We also give a self-contained construction of parabolic curves, which are integral to this new method.

Keywords
Fatou set, holomorphic dynamics, parabolic implosion, polynomial mappings, skew-products, wandering Fatou components, parabolic curves, nonautonomous dynamics
Mathematical Subject Classification 2010
Primary: 32H50, 37F10, 37G15
Milestones
Received: 14 October 2019
Revised: 23 February 2021
Accepted: 14 June 2021
Published: 14 April 2023
Authors
Matthieu Astorg
Institut Denis Poisson
Collegium Sciences et Techniques
Université d’Orléans
Orléans
France
Luka Boc Thaler
Faculty of Education
University of Ljubljana
Ljubljana
Slovenia
Institute of Mathematics, Physics and Mechanics
Ljubljana
Slovenia
Han Peters
Korteweg de Vries Institute for Mathematics
University of Amsterdam
Amsterdam
Netherlands

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