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Jinsong Zeng

Publications and source records attributed to Jinsong Zeng.

17 recordsLinked to original sources
arXiv2026

Non-uniqueness of characteristic curves of folding rational maps

A folding rational map is a postcritically finite rational map $f$ admitting an essential Jordan curve $β$, disjoint from the postcritical set $P_f$, whose preimage is disconnected and consists of Jordan curves homotopic to $β$ relative to $P_f$. Such $β$ is called a characteristic curve of $f$. We construct a real postcritically finite rational map $f$ of degree $6$ with $\#P_f=4$, which admits two non-homotopic characteristic curves $β$ and $β'$. It is hyperbolic with orbifold signature $(2,2,3,\infty)$. This non-uniqueness is not a Lattès phenomenon, nor does it arise from a mating decomposition. The map is constructed piecewise, and a numerical model is provided to illustrate its dynamics. Thurston obstructions are excluded by arc-lifting and intersection-number arguments.

math.DS↗
arXiv2026

Rigidity of McMullen Julia sets

We provide a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps $f_λ(z)=z^n+λ/z^n$ with $λ\in\mathbb{C}^*$ and $n\geq 2$, and prove that the quasisymmetry group of each such Julia set is exactly the finite dihedral group generated by the natural symmetries of the map. These results establish quasisymmetric rigidity for all topological classes in this family, including Sierpiński-like carpets, necklaces, and clusters, and provide the first known examples of rigid Julia sets in each of the three classes.

math.DS↗
arXiv2026

Polynomials in molecules

This paper characterizes polynomials within molecules. We show that a geometrically finite polynomial of degree $d\geq2$ lies in a molecule if and only if all its critical points belong to maximal Fatou chains, and show that distinct molecules are mutually disjoint. We also establish a necessary and sufficient condition for subhyperbolic polynomials to be on the closures of bounded hyperbolic components.

math.DS↗
arXiv2025

Most Fatou and Julia components are small for polynomials

We prove that Julia components of polynomials are generally small in diameter. For polynomials without irrationally neutral cycles, Fatou components are also typically small, even when the Julia set is not locally connected.

math.DS↗
arXiv2024

Invariant graphs in Julia sets and decompositions of rational maps

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\overline{\mathbb{C}}$, and for each sufficiently large integer $n$, there exists a finite and connected graph $G$ in the Julia set of $f$ such that $f^n(G) \subset G$. This graph contains all post-critical points in the Julia set, while every component of $\overline{\mathbb{C}}\setminus G$ contains at most one post-critical point in the Fatou set. The proof relies on the cluster-Sierpinski decomposition of post-critically finite rational maps.

math.DS↗
arXiv2023

Rigidity of non-renormalizable Newton maps

Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries no invariant line fields and that the topological conjugacy is equivalent to quasi-conformal conjugacy in this case.

math.DS↗
arXiv2022

Invariant graphs of rational maps

Let $f$ be a postcritically finite rational map. We prove that, as $n$ large enough, there exists an $f^n$-invariant (finite connected) graph on $\widehat{\mathbb{C}}$ such that it contains the postcritical set of $f$.

math.DS↗
arXiv2020

Criterion for rays landing together

We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set.

math.DS↗
arXiv2020

On rational maps with buried critical points

In this paper, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components by using the pinching and plumbing deformations.

math.DS↗
arXiv2018

Dynamics of Newton maps

In this paper, we study the dynamics of Newton maps for arbitrary polynomials. Let $p$ be an arbitrary polynomial with at least three distinct roots, and $f$ be its Newton map. It is shown that the boundary $\partial B$ of any immediate root basin $B$ of $f$ is locally connected. Moreover, $\partial B$ is a Jordan curve if and only if ${\rm deg}(f|_B)=2$. This implies that the boundaries of all components of root basins, for all polynomials' Newton maps, from the viewpoint of topology, are tame.

math.DS↗
arXiv2018

Quasisymmetric geometry of Sierpinski carpet Julia sets

In this paper, the main focus is on the Sierpinski carpet Julia sets of the rational maps with non-recurrent critical points. We study the uniform quasicircle property of the peripheral circles, the relatively separated property of the peripheral circles and the locally porous property of these carpets. We also establish some quasisymmetric rigidities of these carpets, which generalizes the main results of Bonk-Lyubich-Merenkov to the postcritically infinite case. In the end we give a strategy to construct a class of postcritically infinite rational maps whose Julia sets are quasisymetrically equivalent to some round carpets.

math.DS↗
arXiv2015

Non-recurrent parameter rays of the Mandelbrot set

In this paper, we prove that any parameter ray at a non-recurrent angle $θ$ lands at a non-recurrent parameter $c$ with $θ$ a characteristic angle of $f_c$; and conversely, every non-recurrent parameter $c$ is the landing point of one or two parameter rays at non-recurrent angles, and these angles are exactly the characteristic angles of $f_c$.

math.DS↗
arXiv2015

A characterization of Sierpinski carpet Rational maps

In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet

math.DS↗
arXiv2015

Invariant Jordan curves of Sierpiski carpet rational maps

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $Γ$ containing the postcritical set of $R$.

math.DS↗
arXiv2013

On the dynamics of a family of renormalization transformations

We study the family of renormalization transformations of the generalized $d$--dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where $d\geq 2$. In particular, we prove that their Julia sets can never be a Sierpiński carpet if the parameter is real. We show that the Julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models. Moreover, the asymptotic formula of the Hausdorff dimension of the Julia set is calculated as the parameter tends to infinity.

math.DS↗
arXiv2013

Quasisymmetric rigidity of Sierpinski carpets $F_{n,p}$

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group. We also establish that $F_{n,p}$ and $F_{n',p'}$ are quasisymmetrically equivalent if and only if $(n,p)=(n',p')$.

math.CV↗