通过稳定多重曲线对有理映射的分解

Tame rational functions: Decompositions of iterates and orbit intersections

Let $A$ be a rational function of degree at least two on the Riemann sphere. We say that $A$ is tame if the algebraic curve $A(x)-A(y)=0$ has no factors of genus zero or one distinct from the diagonal. In this paper, we show that if tame rational functions $A$ and $B$ have orbits with infinite intersection, then $A$ and $B$ have a common iterate. We also show that for a tame rational function $A$ decompositions of its iterates $A^{\circ d},$ $d\geq 1,$ into compositions of rational functions can be obtained from decompositions of a single iterate $A^{\circ N}$ for $N$ big enough.

A canonical decomposition of postcritically finite rational maps and their maximal expanding quotients

We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou components are Jordan disks with disjoint closures (Sierpiński maps), as well as those where any two Fatou components can be connected through a countable chain of Fatou components with common boundary points (crochet or Newton-like maps). We provide several alternative characterizations for our decomposition, as well as an algorithm for its effective computation. …

Julia 集中的不变图与有理映射的分解

Invariant graphs in Julia sets and decompositions of rational maps

本文证明任意后临界有限有理映射的 Julia 集中存在在迭代下不变且包含全部后临界点的有限连通图，其构造基于 cluster-Sierpinski 分解。

On decomposable rational maps

If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.

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 finit…

实数和复数上有理函数的分解及一个关于不变曲线的问题

Decompositions of rational functions over real and complex numbers and a question about invariant curves

本文研究实数与复数上有理函数的函数分解之间的联系，并探讨黎曼球面上在有理函数作用下保持不变的曲线问题。

平面紧集与有理Julia集的一个模型

A model for planar compacta and rational Julia sets

该研究提出平面紧集的具有Peano商的核心分解与原子概念，证明分支覆盖下紧集原子的原像分支仍为原像紧集的原子，将多项式下完全不变紧集的相关结果推广到有理函数情形。

Geometrization of postcritically finite branched coverings

We study canonical decompositions of postcritically finite branched coverings of the 2-sphere, as defined by K.~Pilgrim. We show that every hyperbolic cycle in the decomposition does not have a Thurston obstruction. It is thus Thurston equivalent to a rational map.

Geometrization of postcritically finite branched coverings (revised)

We study canonical decompositions of postcritically finite branched coverings of the 2-sphere, as defined by K. Pilgrim. We show that every hyperbolic cycle in the decomposition does not have a Thurston obstruction. It is thus Thurston equivalent to a rational map.

离散动力学的有理解释器：$p$-adic域上的存在性、精确性与分解

Rational Interpreters for Discrete Dynamics: Existence, Exactness, and Decomposition over $p$-adic Fields

本文研究了$p$-adic域上离散动力学的有理解释器的存在性、精确性与分解，提出动态中国剩余定理并探讨了逆极限扩展。