Iteration of Quasiconformal Maps

Quasiconformal maps provide a powerful tool for complex analysis and complicated dynamics studies. In this article, the iterations of quasiconformal maps are investigated, showing that the iterative dynamics can be very complicated. One kind of quasiconformal map in the sense of Beltrami has a global attractor and a growing horseshoe, where the latter means that the number of folds of the horseshoe is increasing as a parameter is varied. Further, the generalized Hénon maps are represented as generalized quasiconformal maps in certain parameter regions. It is shown that this class of generalized quasiconformal maps is different from the maps in the sense of Beltrami or of Beltrami-David. Moreover, a natural class of invariant measures are constructed for the generalized Hénon maps. Based on the characteristics of the quasiconformal maps, some new properties of the generalized Hénon maps are revealed and analyzed, which are related to the existence of the wandering domains. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.