arXiv subjects

Dierk Schleicher

Publications and source records attributed to Dierk Schleicher.

At least 19 recordsLinked to original sources

arXiv2026

Filament Pairs, Dynamic Partitions, and Spiders for Post-Singularly Finite Entire Functions

Filaments are a natural generalization of the well-known concept of dynamic rays in complex dynamics. In this article we investigate which periodic or preperiodic filaments land together for arbitrary post-singularly finite transcendental entire functions. Our first main result is a combinatorial description of the landing relation of filaments in terms of the dynamic partitions of the space of external addresses. One of the main difficulties deals with taming the more complicated topology of filaments. In the end, filaments possess all the topological properties of dynamic rays that are essential for the construction of dynamic partitions. The results of this paper are the foundation for the development of combinatorial models, in particular homotopy Hubbard trees, for arbitrary post-singularly finite transcendental entire functions. Our second mail result is that every postsingularly finite entire function has an iterated that possesses and invariant spider: spiders are, like homotopy Hubbard trees, an important tool for the combinatorial classification of postsingularly finite polynomials and presumably also for entire functions.

math.DS↗

arXiv2025

Diophantine Equations over $\mathbb Z$: Universal Bounds and Parallel Formalization

This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the decidability of Diophantine equations and has been answered negatively by Davis, Putnam, Robinson and Matiyasevich. It is natural to ask for which subclasses of Diophantine equations H10 remains undecidable. Such subclasses can be defined in terms of universal pairs: bounds on the number of variables $ν$ and degree $δ$ such that all Diophantine equations can be rewritten in at most this complexity. Our work develops explicit universal pairs $(ν, δ)$ for integer unknowns, achieving new bounds that cannot be obtained by naive translations from known results over $\mathbb N$. In parallel, we have conducted a formal verification of our results using the proof assistant Isabelle. While formal proof verification has traditionally been applied a posteriori to known results, this project integrates formalization into the discovery and development process. In a final section, we describe key insights gained from this unusual approach and its implications for mathematical practice. Our work contributes both to the study of Diophantine equations and to the broader question of how mathematics is conducted in the 21st century.

math.NT↗

arXiv2024

Hölder continuity of core entropy for non-recurrent quadratic polynomials

We prove that core entropy is Hölder continuous as a function of external angles for a large class of quadratic polynomials that are non-recurrent with respect to angle-doubling, in particular all of them that exhibit a finite Hubbard tree. The result follows from a symbolic analysis of the Mandelbrot set and the dynamics of Hubbard trees in terms of kneading sequences which has been established in previous work.

math.DS↗

arXiv2024

Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

We construct combinatorial Hubbard trees for all unicritical polynomials, and for all exponential maps, for which the critical (singular) value does not escape. More precisely, out of an external angle, or more generally a kneading sequence, we construct a forward invariant tree for which the critical orbit has the given kneading sequence. When the critical orbit of a unicritical polynomial is periodic or preperiodic, then the existence of a Hubbard tree is classical. Our trees exist even when the critical orbit is infinite (in many cases, this yields infinite trees), and even when the polynomial Julia set fails to be path connected. In the latter case, our trees cannot be reconstructed from the Julia set in complex dynamics. Our trees also exist for kneading sequences that do not arise in complex dynamics (for non complex-admissible kneading sequences).

math.DS↗

arXiv2022

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. We also show that postcritically finite rational maps have dynamically natural quotients in which all crochet maps are collapsed to points, while all Sierpiński maps become small spheres; the quotient is a maximal expanding cactoid. The constructions work in the more general setup of Böttcher expanding maps, which are metric models of postcritically finite rational maps.

math.DS↗

arXiv2022

Mathematical Proof Between Generations

A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e. precision and intuition, through the advent of computer proof assistants. For the most time this has been a topic for experts in specialized communities. However, mathematical proofs have become increasingly sophisticated, stretching the boundaries of what is humanly comprehensible, so that leading mathematicians have asked for formal verification of their proofs. At the same time, major theorems in mathematics have recently been computer-verified by people from outside of these communities, even by beginning students. This article investigates the gap between the different definitions of a proof and possibilities to build bridges. It is written as a polemic or a collage by different members of the communities in mathematics and computer science at different stages of their careers, challenging well-known preconceptions and exploring new perspectives.

math.HO↗

arXiv2022

A classification of postcritically finite Newton maps

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 maps is given in terms of a finite connected graph satisfying certain explicit conditions.

math.DS↗

arXiv2021

Homotopy Hubbard Trees for post-singularly finite exponential maps

We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb{C}$, do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a Homotopy Hubbard Tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a Homotopy Hubbard tree that is unique up to homotopy, and we show that post-singulary finite exponential maps can be classified in terms of Homotopy Hubbard Trees, using a transcendental analogue of Thurston's topological characterization theorem of rational maps.

math.DS↗

arXiv2021

Combinatorial properties of Newton maps

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.

math.DS↗

arXiv2021

Unicritical Laminations

Thurston introduced \emph{invariant (quadratic) laminations} in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle doubling map $σ_2$ on the unit circle $\mathbb{S}^1$ were the Central Strip Lemma, non-existence of wandering polygons, the transitivity of the first return map on vertices of periodic polygons, and the non-crossing of minors of quadratic invariant laminations. We use Thurston's methods to prove similar results for \emph{unicritical} laminations of arbitrary degree $d$ and to show that the set of so-called \emph{minors} of unicritical laminations themselves form a \emph{Unicritical Minor Lamination} $\mathrm{UML}_d$. In the end we verify the \emph{Fatou conjecture} for the unicritical laminations and extend the \emph{Lavaurs algorithm} onto $\mathrm{UML}_d$.

math.DS↗

arXiv2020

Rigidity of Newton dynamics

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 has locally connected Julia set. In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable ``in the same way''. Our main tool is the concept of complex box mappings due to Kozlovski, Shen, van Strien; we also extend a dynamical rigidity result for such mappings so as to include irrationally indifferent or renormalizable situations.

math.DS↗

arXiv2020

Puzzles and the Fatou-Shishikura injection for rational Newton maps

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 (puzzle pieces). This puzzle construction is for rational Newton maps what Yoccoz puzzles are for polynomials: it provides the foundation for all kinds of rigidity results of Newton maps beyond our Fatou-Shishikura injection. Moreover, it gives necessary structure for a classification of the postcritically finite maps in the spirit of Thurston theory.

math.DS↗

arXiv2020

The Weierstrass root finder is not generally convergent

Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root finding method of Weierstrass, a root finder that tries to approximate all roots of a given polynomial in parallel (in the Jacobi version, i.e., with parallel updates). This method has a good reputation for finding all roots in practice except in obvious cases of symmetry, but very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well known Newton method, is not generally convergent: there are open sets of polynomials $p$ of every degree $d \ge 3$ such that the dynamics of the Weierstrass method applied to $p$ exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to $Z^3 + Z + 180$ have attracting cycles of period $4$. Here, period $4$ is minimal: we show that for cubic polynomials, there are no periodic orbits of length $2$ or $3$ that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree $d\ge 3$ there are orbits that are defined for all iterates but converge to $\infty$; this is a problem that does not occur for Newton's method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra.

math.DS↗

arXiv2020

Finding polynomial roots by dynamical systems -- a case study

We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich-Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich-Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.

math.NA↗

arXiv2019

Hausdorff dimension of biaccessible angles for quadratic polynomials

A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called biaccessible if two parameter rays land at $z$. In both cases, we say that the external angles of these two rays are biaccessible as well. In this paper we give upper and lower bounds for the Hausdorff dimension of biaccessible external angles of quadratic polynomials, both in the dynamical and parameter space. In particular, explicitly describe those quadratic polynomials where this dimension equals 1 (if and only if the Julia set is an interval), and when it equals 0, namely, at finite direct bifurcations from the polynomial $z^2$, as well as limit points thereof.

math.DS↗

arXiv2018

A Combinatorial classification of postcritically fixed Newton maps

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.

math.DS↗

arXiv2017

Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$ and $O(d\ln^3 d)$ (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.

math.NA↗

arXiv2017

Newton's method in practice: finding all roots of polynomials of degree one million efficiently

We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm that can be applied routinely to find all roots without deflation and with the inherent numerical stability of Newton's method. We specify an algorithm that provably terminates and finds all roots of any polynomial of arbitrary degree, provided all roots are distinct and exact computation is available. It is known that Newton's method is inherently stable, so computing errors do not accumulate; we provide an exact bound on how much numerical precision is sufficient.

math.NA↗