Renormalization and wandering continua of rational maps

Guizhen Cui (1) , Wenjuan Peng (1) , Lei Tan (2)

1 Institute of Mathematics
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent work of Cui-Tan shows that these maps can be decomposed into postcritically fnite renormalization pieces. The main purpose of the present work is to perform the surgery one step deeper. Based on Thurston's idea of decompositions along multicurves, we introduce a key notion of Cantor multicurves (a stable multicurve generating infnitely many homotopic curves under pullback), and prove that any postcritically fnite piece having a Cantor multicurve can be further decomposed into smaller postcritically fnite renormalization pieces. As a byproduct, we establish the presence of separating wandering continua in the corresponding Julia sets. Contrary to the polynomial case, we exploit tools beyond the category of analytic and quasiconformal maps, such as Rees-Shishikura's semi-conjugacy for topological branched coverings that are Thurston-equivalent to rational maps.

Mots clés: renormalization; wandering continua; Julia sets; multicurves; decompositions

Domaines: Systèmes dynamiques [math.DS]

Guizhen Cui, Wenjuan Peng, Lei Tan. Renormalization and wandering continua of rational maps. 2011. ⟨hal-00601632⟩