Searcharxiv⌕ Search

arXiv subjects

Dierk Schleicher

Publications and source records attributed to Dierk Schleicher.

54 records · Page 3Linked to original sources
arXiv2007

Bifurcations in the Space of Exponential Maps

This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+κ$. We prove that the boundary (in $\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in $\C$, which is an analog of Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected. We show furthermore that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demonstrating a general "Squeezing Lemma", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.

math.DS↗
arXiv2007

Hausdorff dimension, its properties, and its surprises

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set $E\subset\C$ of positive planar measure and with dimension 2 such that each point in $E$ can be joined to $\infty$ by one or several curves in $\C$ such that all curves are disjoint from each other and from $E$, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in $\C$ which is not on one of these curves is in $E$. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as $z\mapsto π\sin(z)$.

math.DS↗
arXiv2007

Fractional Sums and Euler-like Identities

We introduce a natural definition for sums of the form \[ \sum_{ν=1}^x f(ν) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the gamma function or Euler's little-known formula \sum_{ν=1}^{-1/2} \frac 1ν= -2\ln 2. Many classical identities like the geometric series and the binomial theorem nicely extend to this more general setting. Sums with a fractional number of terms are closely related to special functions, in particular the Riemann and Hurwitz zeta functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like \[ \lim_{n\to\infty}[ e^{\frac n 4(4n+1)}n^{-\frac 1 8 - n(n+1)}(2π)^{-\frac n 2} \prod_{k=1}^{2n} Γ(1+\frac k 2)^{k(-1)^k} ] = \sqrt[12]{2} \exp({5/24} - \frac 3 2 ζ'(-1) -\frac{7ζ(3)}{16π^2}) \] some of which seem to be new.

math.CA↗
arXiv2007

A Combinatorial Classification of Postsingularly Finite Complex Exponential Maps

We give a combinatorial classification of postsingularly finite exponential maps in terms of external addresses starting with the entry 0. This is an extension of the classification results for critically preperiodic polynomials \cite{BFH} to exponential maps. Our proof relies on the topological characterization of postsingularly finite exponential maps given recently in \cite{HSS}. Our results illustrate once again the fruitful interplay between combinatorics, topology and complex structure which has often been successful in complex dynamics.

math.DS↗
arXiv2007

Classification of Escaping Exponential Maps

We give a complete classification of the set of parameters $κ$ for which the singular value of $E_κ:z\mapsto \exp(z)+κ$ escapes to infinity under iteration. In particular, we show that every path-connected component of this set is a curve to infinity.

math.DS↗
arXiv2006

On Newton's Method for Entire Functions

The Newton map N_f of an entire function f turns the roots of f into attracting fixed points. Let U be the immediate attracting basin for such a fixed point of N_f. We study the behavior of N_f in a component V of C\U. If V can be surrounded by an invariant curve within U and satisfies the condition that each point in the extended plane has at most finitely many preimages in V, we show that V contains another immediate basin of N_f or a virtual immediate basin. A virtual immediate basin is an unbounded invariant Fatou component in which the dynamics converges to infty through an absorbing set.

math.DS↗
arXiv2005

An Introduction to Conway's Games and Numbers

This is an introduction into John Conway's beautiful Combinatorial Game Theory, providing precise statements and detailed proofs for the fundamental parts of his theory. (1) Combinatorial game theory, (2) the GROUP of games, (3) the FIELD of numbers, (4) ordinal numbers, (5) games and numbers, (6) infinitesimal games, (7) impartial games.

math.CO↗
arXiv2005

Combinatorics of Bifurcations in Exponential Parameter Space

We give a complete combinatorial description of the bifurcation structure in the space of exponential maps $z\mapsto\exp(z)+κ$. This combinatorial structure is the basis for a number of important results about exponential parameter space. These include the fact that every hyperbolic component has connected boundary, a classification of escaping parameters, and the fact that all dynamic and parameter rays at periodic addresses land.

math.DS↗
arXiv2005

Parameter rays for the exponential family

We investigate the set of parameters $κ\in\C$ for which the singular orbit $(0,e^κ,...)$ of $E_κ(z):=\exp(z+κ)$ converges to $\infty$. These parameters are organized in smooth curves in parameter space called \emph{parameter rays}.

math.DS↗
arXiv2004

The Dynamical Fine Structure of Iterated Cosine Maps and a Dimension Paradox

We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called ``rays'', each connecting $\infty$ to a well-defined ``landing point'' in $\C$, so that every point in $\C$ is either on a unique ray or the landing point of finitely many rays. The key features of this paper are the following two: (1) this is the first example of a transcendental dynamical system where the Julia set is all of $\C$ and the dynamics is described in detail using symbolic dynamics; and (2) we get the strongest possible version (in the plane) of the ``dimension paradox'': the set $R$ of rays has Hausdorff dimension 1, and each point in $\C\sm R$ is connected to $\infty$ by one or more disjoint rays in $R$; as a complement of a 1-dimensional set, $\C\sm R$ has of course Hausdorff dimension 2 and full Lebesgue measure.

math.DS↗
arXiv2004

Hyperbolic Components in Exponential Parameter Space

We discuss the space of complex exponential maps $\Ek\colon z\mapsto e^{z}+κ$. We prove that every hyperbolic component $W$ has connected boundary, and there is a conformal isomorphism $Φ_W\colon W\to\half^-$ which extends to a homeomorphism of pairs $Φ_W\colon(\ovl W,W)\to(\ovl\half^-,\half^-)$. This solves a conjecture of Baker and Rippon, and of Eremenko and Lyubich, in the affirmative. We also prove a second conjecture of Eremenko and Lyubich.

math.DS↗
arXiv2004

Immediate and Virtual Basins of Newton's Method for Entire Functions

We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce ``virtual immediate basins'' in which the dynamics converges to infinity; we prove that these are simply connected as well.

math.DS↗
arXiv2004

Escaping Points of the Cosine Family

We study the dynamics of iterated cosine maps $E\colon z \mapsto ae^z+be^{-z},$ with $a,b \in \C\setminus \{0\}$. We show that the points which converge to infinity under iteration are organized in the form of rays and, as in the exponential family, every escaping point is either on one of these rays or the landing point of a unique ray. Thus we get a complete classification of the escaping points of the cosine family, confirming a conjecture of Eremenko in this case. We also get a particularly strong version of the ``dimension paradox'': the set of rays has Hausdorff dimension 1, while the set of points these rays land at has not only Hausdorff dimension 2 but infinite planar Lebesgue measure.

math.DS↗
arXiv1999

On Fibers and Local Connectivity of Compact Sets in C

A frequent problem in holomorphic dynamics is to prove local connectivity of Julia sets and of many points of the Mandelbrot set; local connectivity has many interesting implications. The intention of this paper is to present a new point of view for this problem: we introduce fibers of these sets, and the goal becomes to show that fibers are ``trivial'', i.e. they consist of single points. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles. This implies local connectivity at these points, but triviality of fibers is a somewhat stronger property than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful. Since we believe that fibers may be useful in further situations, we discuss their properties for arbitrary compact connected and full sets in the complex plane. This allows to use them for connected filled-in Julia sets of polynomials, and polynomials of the form $z^d+c$ have the property that the impression of any dynamic ray at a rational angle is a single point. An appendix reviews known topological properties of compact, connected and full sets in the plane. The definition of fibers grew out of a new brief proof that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. This proof works also for ``Multibrot sets'', which are the higher degree cousins of the Mandelbrot set. These sets are discussed in a self-contained sequel (IMS Preprint 1998/13a). Finally, we relate triviality of fibers to tuning and renormalization in IMS Preprint 1998/13b.

math.DS↗
arXiv1999

On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets

We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set and show that fibers of certain points are ``trivial'', i.e., they consist of single points. This implies local connectivity at these points. Locally, triviality of fibers is strictly stronger than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful. We include the proof that local connectivity of the Mandelbrot set implies density of hyperbolicity in the space of quadratic polynomials. We write our proofs more generally for Multibrot sets, which are the loci of connected Julia sets for polynomials of the form $z\mapsto z^d+c$. Although this paper is a continuation of preprint 1998/12, it has been written so as to be independent of the discussion of fibers of general compact connected and full sets in $\C$ given there.

math.DS↗
arXiv1999

On Fibers and Renormalization of Julia Sets and Multibrot Sets

We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of fibers of Mandelbrot and Multibrot sets is preserved under tuning maps and other (partial) homeomorphisms. Similarly, we show for unicritical polynomials that triviality of fibers of Julia sets is preserved under renormalization and other surgery procedures, such as the Branner-Douady homeomorphisms. We conclude with various applications about quadratic polynomials and its parameter space: we identify embedded paths within the Mandelbrot set, and we show that Petersen's theorem about quadratic Julia sets with Siegel disks of bounded type generalizes from period one to arbitrary periods so that they all have trivial fibers and are thus locally connected.

math.DS↗
arXiv1998

Rational parameter rays of the Mandelbrot set

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set. Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.

math.DS↗