arXiv subjects

Wenjuan Peng

Publications and source records attributed to Wenjuan Peng.

8 recordsLinked to original sources

arXiv2026

On the parabolic Fatou domains II: rigidity

This paper is a follow-up study on the holomorphic model problem for infinitely-connected parabolic Fatou domains of rational maps. We prove that simple parabolic maps serve as holomorphic models for such parabolic Fatou domains. Moreover, we show that every simple parabolic map can be perturbed into a rational map with a completely invariant attracting Fatou domain without changing the topology of the Julia set, thereby confirming the Goldberg--Milnor conjecture for simple parabolic maps.

math.DS↗

arXiv2025

On the parabolic Fatou domains

Let $f$ be a rational map with an infinitely-connected fixed parabolic Fatou domain $U$. We prove that there exists a rational map $g$ with a completely invariant parabolic Fatou domain $V$, such that $(f,U)$ and $(g,V)$ are conformally conjugate, and each non-singleton Julia component of $g$ is a Jordan curve which bounds a superattracting Fatou domain of $g$ containing at most one postcritical point. Furthermore, we show that if the Julia set of $f$ is a Cantor set, then the parabolic Fatou domain can be perturbed into an attracting one without affecting the topology of the Julia set.

math.DS↗

arXiv2024

On the boundary of the central quadratic hyperbolic component

We give a concrete description for the boundary of the central quadratic hyperbolic component. The connectedness of the Julia sets of the boundary maps are also considered.

math.DS↗

arXiv2023

Wandering Julia components of cubic rational maps

We prove that every wandering Julia component of cubic rational maps eventually has at most two complementary components.

math.DS↗

arXiv2020

On the cycles of components of disconnected Julia sets

For any integers $d\ge 3$ and $n\ge 1$, we construct a hyperbolic rational map of degree $d$ such that it has $n$ cycles of the connected components of its Julia set except single points and Jordan curves.

math.DS↗

arXiv2015

Renormalization and wandering continua of rational maps

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.

math.DS↗

arXiv2015

Renormalizations and wandering Jordan curves of rational maps

We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of uncountably many Jordan curve components. Most of them are wandering. The other consists of components that are pullbacks of finitely many renormalizations, together with possibly uncountably many points. The quotient action on the decomposed pieces is encoded by a dendrite dynamical system. We also introduce a surgery procedure to produce post-critically finite rational maps with wandering Jordan curves and prescribed renormalizations.

math.DS↗

arXiv2011

Combinatorial rigidity of multicritical maps

We combine the KSS nest constructed by Kozlovski, Shen and van Strien, and the analytic method proposed by Avila, Kahn, Lyubich and Shen to prove the combinatorial rigidity of multicritical maps.

math.DS↗