Alexandre Eremenko, Sebastian van Strien
Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle.
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Given a polynomial or a rational map f we associate to it a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier ρ\not=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function ρin the space associated to f. The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the…
Independence of multipliers via degenerations of rational maps
本文通过有理映射的退化方法，证明次数为$d$的有理映射中任意$2d-2$个不同周期轨道的乘子代数独立，条件为至多$d$个不动点，并改进了早期结果。
In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.
Multiplier scales of a sequence of rational maps
该研究分析退化复有理映射序列的乘子行为，揭示周期点乘子要么一致有界要么以共同标度激增，并借助Ahbyankar定理证明非平凡乘子标度最多有2d-2个。
In this article, we prove that every unicritical polynomial map that has only rational multipliers is either a power map or a Chebyshev map. This provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
Blow-up of multipliers in meromorphic families of rational maps
本文研究了黎曼球面上有理映射的单参数全纯退化族中周期循环乘子的爆破现象。
In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.
In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
Entire or rational maps with integer multipliers
本文用微分方程方法替代非阿基米德动力学结果，重新证明了 Ji 和 Xie 关于乘子均为虚二次域整数环元素的有理函数分类定理，并将其推广到整函数情形。
We consider the space of degree $n\ge 2$ rational maps of the Riemann sphere with $k$ distinct marked periodic orbits of given periods. First, we show that this space is irreducible. For $k=2n-2$ and with some mild restrictions on the periods of the marked periodic orbits, we show that the multipliers of these periodic orbits, considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree $n$ rational maps can be locally parameterized by the multipliers of any $2n-2$ …
Entire maps with rational preperiodic points and multipliers
本文证明了对于非实数域的数域，存在预周期点和乘数均位于其中的整映射稠密集，并发现了一个乘数全为有理数且不共轭于仿射映射的逃逸类二次映射。
Generic reconstruction of rational maps from multipliers of periods one and two
本文证明对任意次数d≥2且特征非2的域，模空间上由周期一、二乘子构成的谱态射双有理于其像闭包，从而在代数闭域上通用单射，证实了Ji-Xie猜想，方法涉及正规形、非阿基米德退化与对不变量重构。
We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb C(z)$. For a fixed $K \gg 1$, we show that almost all intervals of length $2π/K$ in $(-π,π]$ contain a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in $K$.
Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics
本文利用复动力学与非阿基米德动力学工具，给出 McMullen 乘子谱刚性定理的新证明，并将其推广到长度谱情形，同时解决 Milnor 关于例外映射的两个猜想。
Dynamical Systems of Correspondences on the Projective Line II: Degrees of Multiplier Maps
本文研究乘子反问题，给出等谱对应数量的显式上界，并以两种计算方法证明 Hutz-Tepper 关于一般三次态射乘子唯一性结果的勘误修正。
New Normal Forms For Degree Three Polynomials and Rational Functions
针对动力系统模空间中共轭类代表选取的有理性难题，提出不动点集与乘子集相等的三次多项式范式，并分类具备类似性质的三次有理映射。
Let $Λ$ be a quasi-projective variety and assume that, either $Λ$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $Λ$ parametrizes an algebraic family $(f_λ)_{λ\inΛ}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof…
We establish a quantitative approximation formula of the Lyapunov exponent of a rational function of degree more than one over an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and possibly non-archimedean absolute value, in terms of the multipliers of periodic points of the rational function. This quantifies both our former convergence result over general fields and the one-dimensional version of Berteloot--Dupont--Molino's one over archimedean fields.
Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-…
We prove that there is no entire transcendental function in class S with real multipliers of all repelling periodic orbits.
Newton's method applied to rational functions: Fixed points and Julia sets
本文研究了牛顿映射在有理函数中的固定点与朱利亚集性质，证明了特定条件下朱利亚集的连通性和局部连通性。
We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > δ- ε$, where $…
Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $φ(z) \in K(z)$ have degree two. We describe the crucial set of $φ$ in terms of the multipliers of $φ$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $φ$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set.
Emergence of Mandelbrot-like and Julia-like Structures in Parameter Slices of Rational Maps
研究有理映射参数切片中的分形结构，揭示其与多项式家族的相似性及动态特性。
Space spanned by characteristic exponents
论文证明了乘积谱和长度谱的刚性结果，展示非例外有理映射的特征指数生成的有理向量空间有无限维，推广Levy和Tucker的问题，并用长度谱刻画后临界有限映射，还证明了射影空间上内射的Zariski稠密轨道猜想。
We consider linear Hamiltonian equations in $\mathbb{R}^{4}$ of the following type \begin{equation} \frac{\mathrm{d}γ}{\mathrm{d}t}(t)=J_{4}A(t)γ(t), γ(0)\in\operatorname{Sp}(4,\mathbb{R}), \end{equation} where $J=J_{4}\overset{\text{def}}{=}\begin{bmatrix}0 & \operatorname{Id}_2\\-\operatorname{Id}_2 & 0\end{bmatrix}$ and $A:t\mapsto A(t)$ is a $C^1$-continuous curve in the space of $4\times 4$ real matrices which are symmetric. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with multiplicity two.
Intrinsic semistable reduction loci for the iterations of non-archimedean quadratic rational functions and moduli-theoretic applications
研究非阿基米德二次有理函数迭代的半稳定内在约简域，利用超几何结果函数的约简理论斜率公式计算，并建立与非阿基米德多项式动力学相似的精确站稳性。
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