All bounded type Siegel disks of rational maps are quasi-disks

Renormalization and Siegel disks for complex Hénon maps

We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.

Structural Instability of Semi-Siegel Hénon maps

We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_δ$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.

Siegel disks and renormalization fixed points

In this note we construct hyperbolic fixed points for cylinder renormalization of maps with Siegel disks.

三次 Siegel 多项式与分支测度

Cubic Siegel polynomials and the bifurcation measure

本文证明具有固定 Siegel 圆盘且临界轨道最终落入其中的三次多项式位于分支测度支集中，肯定回答 Dujardin 的问题，并推出支集中存在全纯圆盘、刚性参数集非闭。

由具有 Siegel 盘的 Lavaurs 映射产生的游荡域

Wandering domains arising from Lavaurs maps with Siegel disks

本文构造了由带 Siegel 型不动点的 Lavaurs 映射产生的新型高维游荡域，并通过抛物曲线与精确收敛速率估计证明捕获域存在的充要条件。

Golden mean Siegel disk universality and renormalization

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like ma…

Mating the Basilica with a Siegel Disk

Consider a quadratic polynomial with a fixed Siegel disc of bounded type. Using an adaptation of complex a priori bounds for critical circle maps, we prove that this Siegel polynomial is conformally mateable with the basilica polynomial.

Cylinder renormalization for Siegel disks and a constructive Measurable Riemann Mapping Theorem

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization - a novel type of renormalization for holomorphic maps with a Siegel disk w…

Mating Siegel Quadratic Polynomials

Let F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers theta and nu. Using a new degree 3 Blaschke product model for the dynamics of F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that F can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers theta and nu.

Semi-continuity of Siegel disks under parabolic implosion

We transfer the lower semi-continuity of Siegel disks with fixed Brjuno type rotation numbers to geometric limits. Here, we restrict to Lavaurs maps associated to quadratic polynomials with a parabolic fixed point.

Siegel变换的可积性及一项应用

Integrability of Siegel transforms and an application

本文建立了半单代数$\mathbb{Q}$-群有理表示场景下推广Siegel变换的$L^p$可积性精确代数准则，并结合有效等分布估计推导了秩一旗簇上实点有理逼近个数的有效渐近公式。

Relatively compact Siegel disks with non-locally connected boundaries

We construct holomorphic maps with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a subset of C.

Smooth Siegel disks everywhere

We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at $0$, to be parameterized by the rotation number $α$, to depend on $α$ in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.

The boundaries of golden-mean Siegel disks in the complex quadratic Hénon family are not smooth

As was recently shown by the first author and others, golden-mean Siegel disks of sufficiently dissipative complex quadratic Hénon maps are bounded by topological circles. In this paper we investigate the geometric properties of such curves, and demonstrate that they cannot be $C^1$-smooth.

On the Siegel-Sternberg linearization theorem

We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode's and pde's. This will open up new directions in \textsc{kam}-theory and other applications of ultradifferentiable functions.

On Rational $P$-Adic Dyanamical Systems

In the paper we investigate the behavior of trajectory of rational $p$-adic dynamical system in complex $p$-adic filed $\C_p$. It is studied Siegel disks and attractors of such dynamical systems. We show that Siegel disks may either coincide or disjoin for different fixed points of the dynamical system. Besides, we find the basin of the attractor of the system. It is proved that such kind of dynamical system is not ergodic on a unit sphere with respect to the Haar measure.

有界型三次Siegel多项式的刚性

Rigidity of bounded type cubic Siegel polynomials

本文证明有界型旋转数的不可重整化三次Siegel多项式的组合等价蕴含共形等价，并由此验证了Blokh-Oversteegen-Ptacek-Timorin关于Zakeri单参数切片中不可重整化映射轨迹拓扑结构的猜想。

On the natural extension of a map with a Siegel or Cremer point

In this note we show that the regular part of the natural extension (in the sense of Lyubich and Minsky) of quadratic map $f(z) = e^{2 πi θ}z + z^2$ with irrational $θ$ of bounded type has only parabolic leaves except the invariant lift of the Siegel disk. We also show that though the natural extension of a rational function with a Cremer fixed point has a continuum of irregular points, it can not supply enough singularity to apply the Gross star theorem to find hyperbolic leaves.

Boundaries of Siegel disks - numerical studies of their dynamics and regularity

Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there …

一般轨道的算术次数的存在性与动力 Lang–Siegel 问题

Existence of arithmetic degrees for generic orbits and dynamical Lang-Siegel problem

本文证明支配有理自映射在一般轨道点处算术次数存在，并推广至拟射影簇上的 étale 态射，同时将结果用于动力 Lang–Siegel 问题，刻画局部高度增长与轨道子集的 Banach 密度。

A separation theorem for entire transcendental maps

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their l…

具有光滑退化 Herman 环的有理映射

Rational maps with smooth degenerate Herman rings

本文利用 Avila–Buff–Chéritat 的光滑 Siegel 盘构造及连续的 Siegel 到 Herman 拟共形手术，证明存在带光滑退化 Herman 环的有理映射，并得到正面积无处稠密 Julia 集的不可重整化例子。

Effective rational approximation on spheres

We prove an effective estimate for the counting function of Diophantine approximants on the sphere S$^n$. We use homogeneous dynamics on the space of orthogonal lattices, in particular effective equidistribution results and non-divergence estimates for the Siegel transform, developping on recent results of Alam-Ghosh and Kleinbock-Merrill.

Area and Hausdorff dimension of Sierpiński carpet Julia sets

We prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $s\in(1,2)$, we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly $s$.

Bifurcation currents in holomorphic families of rational maps

The aim of these lectures is the study of bifurcations within holomorphic families of polynomials or rational maps by mean of ergodic and pluripotential theoretic tools.

有界类型塞格尔多项式的刚性

Rigidity of bounded-type Siegel polynomials

本文研究了具有有界类型塞格尔盘的多项式刚性，证明了其朱利亚集的局部连通性和拟共形刚性，并验证了组合刚性猜想。

Amenability and measure of maximal entropy for semigroups of rational maps

In this article we discuss relations between algebraic and dynamical properties of non-cyclic semigroups of rational maps.

On fixed points of rational self-maps of complex projective plane

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the set of fixed point free rational self-maps of $\pp^2$ is closed (modulo "degenerate" maps) in some natural topology on the space of rational self-maps of $\pp^2$.

Thurston type Theorem for sub-hyperbolic rational maps

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurston's Theorem.

Bicritical rational maps with a common iterate

Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map which shares an iterate with $f$ then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.

Rational maps with bad reduction and domains of quasiperiodicity

Consider a rational map $R$ of degree $d\geq 2$ with coefficients over the non-archimedean field $\mathbb{C}_p$, with $p$ a fixed prime number. If $R$ has a cycle of Siegel disks and has good reduction, then it was shown by Rivera-Letelier in his PhD dissertation that a new rational map $Q$ can be constructed from $R$, in such a way that $Q$ will exhibit a cycle of $m$-Herman rings. In this paper, we address the case of rational maps with bad reduction and provide an extension of Rivera-Letelier's result for these class of maps.

On the scaling ratios for Siegel disks

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling ratio. The restriction of the dynamics of the quadratic polynomial to the boundary of the Siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the…

On the representation of maps by Lie transforms

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Scrhöder--Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension.

On the Size of Quadratic Siegel Disks: Part I

If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2iπ\a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a …

Julia集为广义Sierpiński垫圈的有理映射

Rational maps whose Julia sets are generalized Sierpiński gaskets

本文证明了包含Siegel盘、Cremer点或无穷次重正规化的有理映射也能产生广义Sierpiński垫圈Julia集，由此确立了正面积垫圈Julia集的存在性，并给出了判定其存在性的准则。

A bound on the number of rationally invisible repelling orbits

We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are $q<\infty$ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by $q$. In particular, there are at most $q$ rationally invisible repelling periodic orbits. The…

The Fine Structure of Herman Rings

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting.

A decomposition theorem for Herman maps

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed `Herman map' is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result gives an answer to a problem of McMullen in a sense and enables us to prove a Thurston-type theorem for …

The action on cohomology by compositions of rational maps

We use intuitive results from algebraic topology and intersection theory to clarify the pullback action on cohomology by compositions of rational maps. We use these techniques to prove a simple sufficient criterion for functoriality of a composition of two rational maps on all degrees of cohomology and we then reprove the criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony. We conclude with a cautionary example.

Rational maps with rational multipliers

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.

The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψand a unique $\exp(¶(ψ)-ψ)$-conformal measure, where P(ψ) denotes the topological pressure of ψ.

Rationality of the zeta function for Ruelle-expanding maps

We will prove that the zeta function for Ruelle-expanding maps is rational.

On Rational Maps with Two Critical Points

This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points. (originally titled ``On Bicritical Rational Maps'' in September 1997; revised and retitled April 1999)

Rational Misiurewicz maps are rare

We show that the set of Misiurewicz maps has Lebesgue measure zero in the parameter space of rational maps for any fixed degree greater than or equal to 2.

Bounded hyperbolic components of bicritical rational maps

We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.

Puiseux series dynamics of Quadratic rational maps

We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.

Lyapunov spectrum for rational maps

We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere.

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.

二次有理映射周期曲线中横截性的渐近性

Asymptotics of Transversality in Periodic Curves of Quadratic Rational Maps

本文计算了具有周期标记临界点的二次有理映射模空间的欧拉特征数，从而研究其横截性的渐近行为。

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.

Rational Misiurewicz maps for which the Julia set is not the whole sphere

We show that Misiurewicz maps for which the Julia set is not the whole sphere are Lebesgue density points of hyperbolic maps.

有理映射的亚纯族中乘子的爆破

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。

Unicritical polynomial maps with rational multipliers

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.

具有常值Thurston拉回映射的有理映射

Rational maps with constant Thurston pullback mapping

本文研究具有常值Thurston拉回映射的CTP映射，证明所有正则或混合CTP多项式满足McMullen条件，并构造了一类新的CTP映射例子。