arXiv subjects

Dierk Schleicher

Publications and source records attributed to Dierk Schleicher.

At least 37 records · Page 2Linked to original sources

arXiv2017

Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori

In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical points of the Green's function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. They also observed that hyperbolic maps are dense in this family of meromorphic functions in a rather trivial way. In this paper, we study the parameter space of this family of meromorphic functions, which can be written as antiholomorphic perturbations of Weierstrass Zeta functions. On the one hand, we give a complete topological description of the hyperbolic components and their boundaries, and on the other hand, we show that these sets admit natural parametrizations by associated dynamical invariants. This settles a conjecture, made in \cite{LW}, on the topology of the regions in the upper half plane $\mathbb{H}$ where the number of critical points of the Green's function remains constant.

math.DS↗

arXiv2016

On the Efficient Global Dynamics of Newton's Method for Complex Polynomials

We investigate Newton's method as a root finder for complex polynomials of arbitrary degree. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical mathematics, numerics, computer graphics and physics, known methods have either excellent theoretical complexity but cannot be used in practice, or are practically efficient but are a lacking a successful theory behind them. In this manuscript we describe the theoretical complexity of Newton's method for finding all roots of polynomials of given degree and show that it is near-optimal for the known set of starting points that find all roots. This theoretical result is complemented by a recent implementation of Newton's method that finds all roots of various polynomials of degree more than a million, significantly faster than our upper bounds on the complexity indicate, and often much faster than established fast root finders. In some experiments, it was possible to find all roots even with complexity $O(d\log d)$ for degrees exceeding 100 million. Newton's method thus stands out as a method that has merits both from the theoretical and from the practical point of view. Our study is based on the known explicit set of universal starting points that is guaranteed to find all roots of polynomials of degree $d$ (appropriately normalized). We show that this set contains $d$ points that converge very quickly to the $d$ roots: the expected total number of Newton iterations required to find all roots with precision $\eps$ is $O(d^3\log^3d+d\log|\log\eps|)$, which can be further improved to $O(d^2\log^4d+d\log|\log\eps|)$; in the worst case allowing near-multiple roots, the complexity is $O(d^4\log^2d+d^3\log^2d|\log\eps|)$. The arithmetic complexity for all these estimates is the same as the number of Newton iterations steps, up to a factor of $\log^2 d$.

math.DS↗

arXiv2016

Core entropy of quadratic polymonials

We give a combinatorial definition of "core entropy" for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the $α$ fixed point from its negative). This notion extends known definitions that work in cases when the polynomial is postcritically finite or when the topology of the Julia set has good properties, and it applies to all quadratic polynomials in the Mandelbrot set. We prove that core entropy is continuous as a function of the complex parameter. In fact, we model the Julia set as an invariant quadratic lamination in the sense of Thurston: this depends on the external angle of a parameter in the boundary of the Mandelbrot set, and one can define core entropy directly from the angle in combinatorial terms. As such, core entropy is continuous as a function of the external angle. Moreover, we prove a conjecture of Giulio Tiozzo about local and global maxima of core entropy as a function of external angles: local maxima are exactly dyadic angles, and the unique global maximum within any wake occurs at the dyadic angle of lowest denominator. We also describe where local minima occur. An appendix by Wolf Jung relates different concepts of core entropy and biaccessibility dimension and thus shows that biaccessibility dimension is continuous as well.

math.DS↗

arXiv2015

An Extended Fatou-Shishikura inequality and wandering branch continua for polynomials

Let $P$ be a polynomial of degree $d$ with Julia set $J_P$. Let $\widetilde N$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikura inequality $\widetilde N\le d-1$. The goal of the paper is to improve this bound. The new count includes \emph{wandering collections of non-precritical branch continua}, i.e., collections of continua or points $Q_i\subset J_P$ \emph{all} of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of $\mathrm{eval}(Q_i)\ge 3$ external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense. A weak version of the inequality reads \[ \widetilde N+N_{irr}+χ+\sum_i (\mathrm{eval}(Q_i)-2) \le d-1 \] where $N_{irr}$ counts repelling cycles with no periodic rays landing at points in the cycle, $\{Q_i\}$ form a wandering collection $\mathcal{B}_\mathbb{C}$ of non-precritical branch continua, $χ=1$ if $\mathcal{B}_\mathbb{C}$ is non-empty, and $χ=0$ otherwise.

math.DS↗

arXiv2015

On Multicorns and Unicorns II: Bifurcations in Spaces of Antiholomorphic Polynomials

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^d + c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio $2$. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

math.DS↗

arXiv2015

Rational Parameter Rays of The Multibrot Sets

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misiurewicz parameters are landing points of rational parameter rays.

math.DS↗

arXiv2014

On the speed of convergence of Newton's method for complex polynomials

We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\mathcal{S}_d$ of $3.33d\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting points to reach all roots with precision $\varepsilon$ is $O(d^2\log^4 d + d\log|\log \varepsilon|)$. This is an improvement of an earlier result in \cite{Schleicher}, where the number of iterations is shown to be $O(d^4\log^2 d + d^3\log^2d|\log \varepsilon|)$ in the worst case (allowing multiple roots) and $O(d^3\log^2 d(\log d + \log δ) + d\log|\log \varepsilon|)$ for well-separated (so-called $δ$-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than $O(d^2)$ for fixed $\varepsilon$.

math.DS↗

arXiv2012

Multicorns are not Path Connected

The "multicorn" is the connectedness locus of unicritical antiholomorphic polynomials $z\mapsto \bar r{z}^d+c$; the special case $d=2$ was named "tricorn" by Milnor. It appears as a natural local configuration in spaces of real cubic polynomials. We prove that no multicorn for $d\ge 2$ is pathwise connected, confirming a classical prediction based on numerical observations.

math.DS↗

arXiv2012

Internal addresses in the Mandelbrot set and Galois groups of polynomials

We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sections 3 and 4). A simple extension, \emph{angled internal addresses}, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Section~6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Section~7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.

math.DS↗

arXiv2012

Bernoulli measure of complex admissible kneading sequences

Iterated quadratic polynomials give rise to a rich collection of different dynamical systems that are parametrized by a simple complex parameter $c$. The different dynamical features are encoded by the \emph{kneading sequence} which is an infinite sequence over $\{0,\1\}$. Not every such sequence actually occurs in complex dynamics. The set of admissible kneading sequences was described by Milnor and Thurston for real quadratic polynomials, and by the authors in the complex case. We prove that the set of admissible kneading sequences has positive Bernoulli measure within the set of sequences over $\{0,\1\}$.

math.DS↗

arXiv2011

A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

We specify a small set, consisting of $O(d(\log\log d)^2)$ points, that intersects the basins under Newton's method of \emph{all} roots of \emph{all} (suitably normalized) complex polynomials of fixed degrees $d$, with arbitrarily high probability. This set is an efficient and universal \emph{probabilistic} set of starting points to find all roots of polynomials of degree $d$ using Newton's method; the best known \emph{deterministic} set of starting points consists of $\lceil 1.1d(\log d)^2\rceil$ points.

math.DS↗

arXiv2011

Hausdorff dimension and biaccessibility for polynomial Julia sets

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1-dimensional measure.

math.DS↗

arXiv2011

How to Add a Noninteger Number of Terms: From Axioms to New Identities

Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way. We illustrate how fractional sums can be used to derive infinite sum and special functions identities; the corresponding proofs turn out to be particularly simple and intuitive.

math.CA↗

arXiv2010

Homeomorphisms between limbs of the Mandelbrot set

We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994.

math.DS↗

arXiv2009

Dynamic rays of bounded-type entire functions

We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

math.DS↗

arXiv2008

Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is sufficient, namely that every point in the bifurcation locus is the landing point of a parameter ray. For exponential maps, the bifurcation locus is not locally connected. We describe a different conjecture (triviality of fibers) which naturally generalizes the role that local connectivity has for quadratic or unicritical polynomials.

math.DS↗

arXiv2008

Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials

Hubbard trees are invariant trees connecting the points of the critical orbits of postcritically finite polynomials. Douady and Hubbard \cite{Orsay} introduced these trees and showed that they encode the essential information of Julia sets in a combinatorial way. The itinerary of the critical orbit within the Hubbard tree is encoded by a (pre)periodic sequence on $\{\0,\1\}$ called \emph{kneading sequence}. We prove that the kneading sequence completely encodes the Hubbard tree and its dynamics, and we show how to reconstruct the tree and in particular its branch points (together with their periods, their relative posititions, their number of arms and their local dynamics) in terms of the kneading sequence alone. Every kneading sequence gives rise to an abstract Hubbard tree, but not every kneading sequence occurs in real dynamics or in complex dynamics. Milnor and Thurston \cite{MT} classified which kneading sequences occur in real dynamics; we do the same for complex dynamics in terms of a complex \emph{admissibility condition}. This complex admissibility condition fails if and only if the abstract Hubbard tree has a so-called \emph{evil} periodic branch point that is incompatible with local homeomorphic dynamics on the plane.

math.DS↗

arXiv2007

Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints

We investigate the set $I$ of parameters $κ$ for which the singular value of $z\mapsto e^z+κ$ converges to $\infty$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. We show that the parameter rays have Hausdorff dimension 1, while the ray endpoints in $I$ alone have dimension 2. Analogous results were known for dynamical planes of exponential maps; our result shows that this also holds in parameter space.

math.DS↗