Johannes Rückert
We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials and indicate potential for extensions. As our main tool, we show that for a large class of Newton maps that includes all hyperbolic ones, every component of the basin of an attracting fixed point can be connected to infinity through a finite chain of such components.
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We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z) \text{exp}(q(z))$ for $p(z)$, $q(z)$ polynomials and $\text{exp}(z)$, the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.
The paper deals with Newton maps of complex exponential functions and a surgery tool developed by P. Haïssinsky. The concept of "Postcritically minimal" Newton maps of complex exponential functions are introduced, analogous to postcritically finite Newton maps of polynomials. The dynamics preserving mapping is constructed between the space of postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of complex exponential functions.
We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.
The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal of finding a classification of general rational maps is so far elusive. Newton maps (rational maps that arise when applying Newton's method to a polynomial) form a most natural family to be studied from the dynamical perspective. Using Thurston's characterization and rigidity theorem, a complete combinatorial classification of postcritically finite Newton m…
This paper constructs a combinatorial model for all postcritically finite rational maps arising as the Newton's method of a complex polynomial. This model is used in [LMS] to give a combinatorial classification of postcritically finite Newton maps of any degree.
Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps
本文从外部角或揉序列出发，为临界值不逃逸的所有单临界多项式与指数映射构造正向不变的组合 Hubbard 树，将经典构造推广到无限临界轨道、Julia 集非道路连通及非复容许揉序列情形。
In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped by Kiwi and Dujardin-Favre. In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the b…
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. …
Using Thurston's characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a rational map and suppose that the postcritical set $P(f)$ is finite. Let $α$ be an embedded closed arc in the sphere and suppose that $f|α$ is a homeomorphism. Define a branched covering $g$ as follows. Cut the sphere open along $α$. Glue in a closed disc $D$. Map $S^{2} - \Int (D)$ via $f$ and $\Int (D)$ by a homeomorphism to the complement of $f(α)$. We prove theorems …
In this paper, we define the core entropy for postcritically-finite Newton maps and study its continuity within this family. We show that the entropy function is not continuous in this family, which is different from the polynomial case studied by Thurston, Gao, Dudko- Schleicher, Tiozzo [Th+, GT, DS, Ti2], and describe completely the continuity of the entropy function at generic parameters.
In this article we provide a combinatorial sufficient (and conjecturally, necessary) condition (called $α$-symmetry) for the mating of two postcritically finite polynomials in $\mathcal{S}_1$ to be obstructed. To do this, we study the rotation sets associated to the parameter limbs in the connectedness locus of $\mathcal{S}_1$, which allows us to determine when there exist ray classes in the formal mating which contain a closed loop. We give a proof of the necessity of $α$-symmetry for a particular subset of postcritically finite maps in $\mathcal{S}_1$. Many examples are given to illustrate t…
Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion---combinatorial, dynamical, algebraic, and coarse-geometric.
We establish a principle that we call the Fatou-Shishikura injection for Newton maps of polynomials: there is a dynamically natural injection from the set of non-repelling periodic orbits of any Newton map to the set of its critical orbits. This injection obviously implies the classical Fatou-Shishikura inequality, but it is stronger in the sense that every non-repelling periodic orbit has its own critical orbit. Moreover, for every Newton map we associate a forward invariant graph (a puzzle) which provides a dynamically defined partition of the Riemann sphere into closed topological disks (pu…
We extend Thurston's combinatorial criterion for postcritically finite rational maps to a class of rational maps with bounded type Siegel disks. The combinatorial characterization of this class of Siegel rational maps plays a special role in the study of general Siegel rational maps. As one of the applications, we prove that for any quadratic rational map with a bounded type Siegel disk, the boundary of the Siegel disk is a quasi-circle which passes through one or both of the critical points.
We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points…
A Newton graph of order $r( \geqslant 2)$ is a cellularly embedded toroidal graph on $r$ vertices, $2r$ edges and $r$ faces that fulfils certain combinatorial properties (Euler, Hall). The significance of these graphs relies on their role in the study of structurally stable elliptic Newton flows - say $\bar{\bar{\mathcal{N}}} (f)$ - of order $r$, i.e. desingularized continuous versions of Newton's iteration method for finding zeros for an elliptic function $f$ (of order $r$). In previous work we established a representation of these flows in terms of Newton graphs. The present paper results in…
In our previous paper we associated to each non-constant elliptic function $f$ on a torus $T$ a dynamical system, the elliptic Newton flow corresponding to $f$. We characterized the functions for which these flows are structurally stable and showed a genericity result. In the present paper we focus on the classification and representation of these structurally stable flows. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded, graph $\mathcal{G}(f)$ on a torus $T$ with $r$ vertices, 2$r$ edges and $r$ faces that fulfil certain combinatoria…
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