# Christopher Bishop

## Professor, Mathematics SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (516)-632-8274
Dept. Phone: (516)-632-8290
FAX: (516)-632-7631

TuTh 1:00pm to 2:20pm, Physics P-128

I am trying to write some lecture notes to go along with the class. You can find the current version here , but they are incomplete, very rough and have no references yet.

## Introduction to the course

This is a topics course on the iteration of transcendental entire functions. The term `transcendental' means that we will consider entire functions that are not polynomials, and we will emphasize the differences between polynomial dynamics and transcendental dynamics. The term transcendental dynamics is also used to describe the iteration of (non-rational) meromorphic functions, but we will mostly stick to the case of entire functions. Thus a better (but longer) title for the course would be `Dynamics of Transcendental Entire Functions'.

The most obvious diference between polynomials and trancendental entire functions is that infinity is always a super attracting fixed point for polynomials, whereas a trancendental function has an essential singularity at infinity and hence takes every value (except possibly one) in every neighborhood of infinity. Thus the points that escape to infinity under a polynomial form a component of the Fatou set, and the boundary of this component is the Julia set. For transcendental functions escaping points may be in either the Julia set or the Fatou set, but it is still true that the Julia set is the boundary of the escaping set. Moreover, in polynomial dynamics, every escaping point escapes at essentially the same speed, but in transcendental dynamics, the escaping set is broken into numerous different subsets definded in terms of "speed of escape", and a large part of the theory to date deals with properties of the escaping sets and its various subsets. Eremenko's conjecture asks if every connected component of the escaping set is unbounded; this is still open despite intense investigation for many years and numerous related results.

Another difference between the poluynomial and transcendental cases is that the Julia set of a transcendental entire function must have non-trivial connected components and hence the Hausdorff dimension of such a Julia set must ge greater or equal to 1. This is due to Baker. McMullen gave explicit examples of entire functions where the Julia set has positive area and examples where the area is zero, but the Hausdorff dimension is \$2\$. Stallard later showed that every dimension strictly between 1 and 2 can also be attained. Recently I constructed an entire function (given as an infinite product of polynomials) whose Julia set has Hausdorff dimension 1 (in fact it has locally finite linear measure and has packing dimension 1 as well). Thus the Hausdorff dimension of a transcendental Julia set can attain every value between 1 and 2 (including the endpoints). However, all currently known examples have packing dimension equal to 1 or 2; no examples with fractional packing dimension are yet known (but I expect this will change soon).

A singular value of an entire function is either a critical value or an finite asymptotic value (i.e., a finite limiting value of f along some curve tending to infinity, as 0 is an asymptotic value of exp(z) along the negative real axis). Away from its singular set, an entire function acts as a covering map and is easy (easier?) to understand. For polynomials, only finitely many critical values occur and there are no finite asymptotic values. Transendental entire functions with only finitely many singular values (like exp and sine) are in some sense "like polynomials". For example, Sullivan's "no wandering domains" theorem states that for polynomials, every component of the Fatou set is preperiodic. However, Baker showed that there are entire functions that have wandering domains. His examples have unbounded (hence infinite) singular sets. Sullivan's theorem was extended by Eremenko-Lyubich and Goldberg-Keen to entire functions with only finitely many singular values. This left open the case when f has an infinite, but bounded, set of singular values. Recently I showed that such a function can have a wandering domain.

The idea for constructing this wandering domain is a geometric construction of entire functions. Much of transcendental dynamics has been motivated by, and concentrated on, explicit families of funtions such as families of exponential functions, trigonometric families and other functions given by explicit infinite products or integrals. My examples are more geometric in nature, e.g., one can draw a union of curves in the plane satisifying certain simple properties and prove that there is an entire function \$f\$ so that the level set \$\{ |f| = 1\}\$ approximates the picture. We build a piecewise linear map that has the desired geometry and also has bounded conformal distortion (i.e., it is quasiregular) and then use the measurable Riemann mapping theorem is convert it to an actual entire function. The new function has exactly the same singular set as the PL model and in many cases of interest the entire function is a very close approximation to the PL model; close enough to deduce some interesting properties, such as the existence of a wandering domain as mentioned above. I call the method "quasiconformal folding" for reasons that should become apparent if you stick with the course long enough.

As will be evident to those attending the class, I am not an expert on dynamics (either polynomial, rational, entire or meromorphic), but in the last year or so I produced some new examples of entire functions that are interesting from a dynamcial point of view. This course is intended to help me to better learn some of the background to these examples and to attempt to explain some (but certainly not all) the details of my constructions. Roughly there will be three components: 1. Everything from analysis you need to know first 2. Basic results from transcendental dynamics 3. Quasiconformal folding and its applications Each of these components could probably fill a course all on its own, so we skip between them. At the beginning (2) will be the main focus, but I hope top shift towards (3) near the end, and bits of (1) will be crammed in as needed, including a review of the relevant parts of polynomial dynamics (but I expect some of the audience will know this better than I do).

## The first paper

Sur l'iteration des fonctions transcendantes Entires by P. Fatou. 1926 Acta Math vol 47 no 4, pages 337-370.

## Surveys

Dynamics of Entire Functions by Dierk Schleicher. This is a 2008 survey that we will use for the basic structure of the course.

An introduction to Complex Dynamics by Walter Bergweiler. Short (35 page) introduction to meromorphic dynamics.

Iteration of meromorphic functions by Walter Bergweiler. Excellent 1993 survey from the Bulletin of the Americal Math Society.

## Background in analysis

The local growth of power series: a survey of the Wiman-Valiron method by Walter Hayman, Canadian Math. Bull. 17(1974)3 317-358.

On an integral function of an integral function by G. polya, 1926, Journal LMS, 1, 12-15. Estimtes the maximum modulus of a composition of two entire functions in terms of their individual moduli. This result is used in Baker's proof that entire functions do not have unbounded, multiply connected Fatou components (Baker, 1975, AASF vol 1, no 2, 277-283).

On the behavior of meromorphic functions in the neighborhood of an isolated singularity by Olli Lehto and K.I. Vitanen, AASF num 240, 1957 Shows that if f is ameromorphi cfunction with an essential singularity at infity then the spherical gradient grows at least like k|z| for some uniform k >0.

Distribution of values and singularities of analytic functions by Olli Lehto , AASF num 249/3, 1957 Survey of Nevanlinna theory and estimate of spherical derivatives.

Boundary behavior of normal meromorphic functions by Olli Lehto and K.I. Virtanen , Acta. Math., vol 97, 1957 page 47-65. Characterizes normal functions in terms of spherical derivative.

On solutions of the Beltrami equation by Melkana Brakalova and James Jenkins. 1998 J. Anal. Math. vol 76, pages 67-92 Gives solutions of degenerate Beltrami equations (sup mu =1 is allwoed).

On solutions of the Beltrami equation II by Melkana Brakalova and James Jenkins. 2004, Publ. de l'Institute Math. Belgrade, Novelle serie, 75(89) page 3-8. Extends results of previous paper.

On the degenerate Beltrami equation by V. Gutlyanskii, O. Martio, T. Sugawa and M. Vuorinen, 2004 TAMS, vol 357m no 3, pages 875-900.

Arkelian's Approximation Theorem by Jean-Pierre Rosay and walter Rudin. Amer. Math. Monthly, vol 96, 1989, 432-434. Gives quite elementary (1 page) proof of Arkelian's approximation theorem for entire functions from better know Runge and Mergelian approximation theorems.

The hyberbolc metric and geometric function theory Alan Beardon and David Minda 2005, Proc. of the international workshop on QC mappings and their applications, Everything you need to know about the hyperbolic metric and variations of Schwarz's lemma.

Some inequalities for the Poincare metric of plane domains by Toshhiyuki Sugawa and Matti Vuorinen, 2005 Math Z., vol 250, n0 4, 885-906 More about the hyperbolic metric.

A proof of the uniformization theorem for arbitrary planar domains by Yuval Fisher, John H. Hubbard and Ben. S. Wittner, 1988, PAMS, vol 104 no 2 pages 413-418.

## Baker

I.N. Baker by Philip Rippon. Obituary of Noel Baker describing the development of transcendental dynamics and Baker's fundamental contributions.

An entire function which has wandering domains by I.N. Baker. 1976 Journal of the Austrailian Math. Soc. First proof that wandering domains can exist.

The domains of normality of an entire function by I.N. Baker. 1975 AASF 1975, vol 1, 277-283 Proves there are no unbounded, multiply connected Fatou components.

Wandering domains in the iteration of entire functions Baker, 1984, Proc. of LMS More wandering domains for entire functions. wandering domain that iterates to infinity.

## Bergweiler

Fixed points of composite entire and quasiregular maps by Walter Bergweiler. 2005 Uses Alhfors' islands theorem to prove the composition of any two entire functions has infinitely many fixed points.

An entire function with no fixed points and no invariant Baker domai by Walter Bergweiler.

A new proof of the Ahlfors Island theorem by Walter Bergweiler. 1998 J. d'Analyse Mat., vol 76, no 1, 337-345

The Role of the Ahlfors five island theorem in complex dynamics by Walter Bergweiler. 2000, Conform. Geom. and Dynam., 4, 22-34

Bloch's Principle by Walter Bergweiler. 2006, Computational Methods and Function Theory, vol 6, no 1, 77-108

The Role of the Ahlfors five island theorem in complex dynamics Simple proofs of some fundamental properties of the Julia set by Detlef Bargmann, 1999 Erg. Thy. and Dyanm. Systems, 19, mo 3, 553-558. Shows Julia set is non-empty and is closure of repelling points without using Nevanlinna theory.

Entire functions of slow growth whose Julia set coincides with the plane by Walter Bergweiler and Alexandre Eremenko.

On the Hausdorff diemsnion of the Julia set of a regularly growing entire function by Walter Bergweiler and Boguslawa Karpin{\' n}ka. 2009. This proves that if the growth rate a function is sufficiently regualr then the Julia set and escaping set have dimension 2.

The growth rate of an entire function and the Hausdorff dimension of its Julia set by Walter Bergweiler, Boguslawa Karpin{\' n}ka and Gwyneth Stallard. 2008 For functions in the Eremenko-Lyubich class, this gives a lower bound on the Hausdorff dimension of the limit set in terms of the order of growth of the function.

On the Julia set of analytic self-maps of the punctured plane by Walter Bergweiler.

Foundation for an iteration theory of entire quasiregular maps by Walter Bergweiler and Danial A. Nicks. Develops quasiregular analog of iteration of transcendental entire functions.

Omitted values in domains of normaility by Walter Bergweiler and Steffen Rohde. One page paper showing iterate of a fatou component covers some Fatou component except possibly for one point. Due independently to M. Herring

On the uniform perfectness of the boundary of multiply connected wandering domains by Walter Bergweiler and Jian-Hua Zheng .

Multiply connected wandering domains of entire funtions by Walter Bergweiler, Philip Rippon and Gwyneth Stallard. Describes behavior of f on a multiply connected wandering domain: iterates of domain must contain annuli which increasing moduli and f approxmates a monomial on these annuli. In particular, the Julia set can't be uniformly perfect (however, they show any Fatou component is regular for the Dirichlet problem).

A property of the derivative of an entire funxtion by Walter Bergweiler and ALexandre Eremenko. 2011. If f is entire and M is unbounded then f'(f^{-1}(M)) is unbounded.

Escape rate and Hausdorff measure for entire functions by Walter Bergweiler and J{\"o}rn Peter.

Periodic points of entire functions: proof of a conjecture of Baker by Walter Bergweiler .

On the packing dimension of the Julia set and escaping set of an entire function by Walter Bergweiler. Various conditions are given that imply the escaping set has packing dimension 2, e.g., f is bounded on curve tending to infinity or its growth rate satisfies log M(2r,f) > d log M(r,f) for some d >1 and all large r.

## Eremenko

Examples of entire functions with pathological dynamics by Alex Eremenko and Misha Lyubich, Journal of the LMS, 1987. They construct several examples of entire funtions with wandering domains and other unusual features.

Dynamical properties of some classes of entire functions by Alex Eremenko and Misha Lyubich, Annals de l'Institute Fourier, 1992. This describes the structure of functions in what now called the Eremenko-Lyubich class and the Spieser class; entire functions with bounded singular set and finite singular set. We will cover many of the ideas described here.

Dynamics of transcendental meromorphic functions by P. Dominguez, Ann. Acad. Sci. Fenn., 1998 vol 23, 225-250, Proves escaping set is non-empty via Bohr's lemma. Also gives numerous results about iteration of meromorphic functions.

On the iteration of entire functions by Alex Eremenko. Baanch Center publications, 1989 Proves escaping set is non-empty via Wiman-Valiron method.

## Hinkanen

Entire functions with bounded Fatou Components by Aimo Hinkkanen.

## Kisaka and Shishikura

On multiply connected wandering domains of entire functions 2008 by Masashi Kisaka and Mitsuhiro Shishikura. They construct wandering domains of any finite connectivity.

## McMullen

Area and Hausdorff dimension of Julia sets of entire functions by Curt McMullen. TAMS 1987. One of the first (if not the first) paper to compute the dimension of a transcendenta l Julia set. This papers deals with functions of the form f(z) = a exp(z) and g(z) = sin(az+b). Julia sets for the first class have zero area but dimension 2, and the second class has Julia sets of positive area.

## Misirewicz

On iterates of exp(z) by Michak Misirewicz, Erg. Thy. Dyn. SyS 1981 vol 1 103-106 Julia set of exp(z) is the whole plane.

## Rempe (aka Rempe-Gilen}

Dynamical rays of bounded-type entire functions by G{\"u}nter Rottenfusser,Johannes R{\"u}ckert, Lasse Rempe and Dierk Schleicher. 2009. Important paper that constructions functions in the Eremenko-Lyubich class for which the strong eremenko conjecture fails: the escaping set has no non-trivial path connected components.

Rigidty of escaping dynamics for transcendental entire functions by Lasse Rempe. 2009. Proves that two Eremenko-Lyubich functions that are QC equivalent are actually QC conjugate near infinity, i.e., on the set of points whose iterates remain larger than some large value R. If the functions are hyperbolic then they are conjugate on the whole escaping set.

Hausdorff dimensios of escaping sets of transcendental entire functions by Lasse Rempe and Gwyneth Stallard. 2009. If two functions in the Eremenko-Lyubich class are QC equivalent then the escaping sets have the same Hausdorff dimension. This is not necessarily true for the Julia sets.

The escaping set of the exponential by Lasse Rempe. 2008. Shows that the escaping set of exp(z) is a connected set. 4 pages.

## Rippon, Stallard and Rippon and Stallard

Baker domains 2008 y Philip Rippon

Fast escaping points of entire functions 2010 by Philip Rippon and Gwyneth Stallard. Gives numerous properties of the fast escaping sets, including spider web structures and connections to Baker's conjecture. Shows that several alternative definitions of A(f) agree.

Dimensions of Julia sets of meromorphic functions by Phil Rippon and Gwyneth Stallard. Shows that the Julia set of a E-L function always has packing dimension 2 (other example of Stallard show the Hausdorff dimension can be close to 1). In fact, the points that escape as qucikly as possible have packing dimension 2.

Boundaries of escaping Fatou components by Phil Rippon and Gwyneth Stallard. Relates whether a Fatou component escapes to whether a `large' set of its boundary points escape.

Escaping points of entire functions of small growth by Phil Rippon and Gwyneth Stallard. 2008. This paper gives growth conditions that imply the escaping set is connected.

A sharp growth condition for a fast escaping spider's web by Phil Rippon and Gwyneth Stallard. 2012. Gives a growth condition that implies the fast escaping set is a spider web.

On questions of Fatou and Eremneko by Phil Rippon and Gwyneth Stallard. 2005.

On multiply connected wandering domains of a meromorphic function by Phil Rippon and Gwyneth Stallard. 2007.

Slow escaping points of meromorphic functions by Phil Rippon and Gwyneth Stallard. 2008. Any meromorphic function contains points that escape to infinity arbitrarily slowly.

The Hausdorff dimension of Julia sets of Entire functions by Gwyneth Stallard, Erg. Thy. Dyn. Sys., 1991, vol 11, page 769-777. Shows that the Julia set of an entore function can have Hausdorff dimension as close to 1 as we wish.

The Hausdorff dimension of Julia sets of Entire functions II by Gwyneth Stallard. Show that Julia set of a E-L function has dimension strictly bigger than 1.

The Hausdorff dimension of Julia sets of Entire functions III by Gwyneth Stallard. More examples of entire functions whose Julia set has Hausdorff dimension strictly between 1 and 2

The Hausdorff dimension of Julia sets of Entire functions IV by Gwyneth Stallard. For each 1< p < 2 an entire function is given whose Julia set has dimension p (this is the first time that fractional dimensions are explicitly computed; in previous cases only estimates were given to prove values near 1 and 2 could be attained by some function).

Dimensions of Julia sets of transcendental meromorphic functions by Gwyneth Stallard. A 2008 survey about dimensions of Julia sets.

## Sixsmith

A new characterization of the Eremenko-Lyubich class by David Sixsmith. 2012. Characterizes E-L functions in terms of the growth of |zf'(z)/f(z)|.

## Sullivan

Quasiconformal homeomorphisms and Dynamics I: solution of the Fatou-Julia problem on wandering domains by Dennis Sullivan, 1985, Annals of Math, vol 122, no 2, pages 401-418 Rational functions have no wandering domains.

## Urbanski

Geometric rigidity for class S of transcendental meromorphic functions whose Julia sets are Jordan curves by Mariusz Urba{\'n}skil Proc. AMS, 2009. Bowen's diohotomy for class S: Julia set is either a line or has dimension > 1, assuming it is a topological curve.

## Others

A transcendental Julia set of dimension 1 by C. Bishop. In 1975 Baker proved that if f is a transcendental entire function, then Julia set of f contains a continuum and hence has Hausdorff dimension at least 1. McMullen had shown dimension 2 is possible and Stallard showed every value in (1,2] is possible, but the question of whether dimension 1 can be attained has remained open. We will show the answer is yes in the strongest possible way by showing there is a transcendental entire function whose Julia set has locally-finite length (any bounded part has finite length). The packing dimension of this example is also 1, the first known example where the dimension is less than 2 (as far as I know).

Constructing entire functions by quasiconformal folding by C. Bishop. We give a method for constructing transcendental entire functions with good control of both the singular values of \$f\$ and the geometry of the tracts of f. The method consists of first building a quasiregular map by ``gluing together'' copies of the right half-plane that have each been quasiconformally ``folded'' into themselves. The measurable Riemann mapping theorem is then invoked to produce an entire function with similar geometry. As an application we construct a wandering domain in the Eremenko-Lyubich class (an entire function with bounded singular set). We also construct Speiser class functions (finite singular set) with tracts that spiral arbitrarily fast, are strong counterexamples to the area conjecture, or have various dynamical pathologies.

The geometry of bounded type entire functions by C. Bishop. We discuss a method for constructing functions in the Eremenko-Lyubich class B of transcendental functions with bounded singular set whose infinite tracts have prescribed geometry. The result is analogous to, but much simplier than, the result for the Speiser class S proved in the previous preprint. In particular, we show that any union of simply connected tracts that meet certain obvious necessary conditions are equivalent to the tracts of some function in class B, but this is not true for class S.

by .

Send the lecturer (C. Bishop) email at: bishop - at - math.sunysb.edu

Send email to the whole class ((C. Bishop and students) class list