Renormalization and wandering continua of rational maps

TL;DR

This paper advances the understanding of complex rational maps by introducing Cantor multicurves for deeper decomposition of postcritically finite maps and revealing wandering continua in Julia sets.

Contribution

It introduces Cantor multicurves and demonstrates their role in further decomposing rational maps into smaller renormalization pieces, extending prior polynomial results.

Findings

Decomposition of rational maps using Cantor multicurves

Existence of wandering continua in Julia sets

Application of topological tools beyond quasiconformal maps

Abstract

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 finite 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 infinitely many homotopic curves under pullback), and prove that any postcritically finite piece having a Cantor multicurve can be further decomposed into smaller postcritically finite 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.