lucjan emil boettcher- the polish pioneer of holomorphic dynamics

Lucjan Emil Böttcher (1872-1937):the Polish pioneer of holomorphic dynamics

Małgorzata Stawiska-Friedland

Mathematical Reviews/MathSciNet, Ann Arbor, USA

Perception of Science in Central and Eastern Europe 1850-1920,Kraków, September 20-22, 2013

M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 1 / 32

Outline

Holomorphic dynamics;Böttcher’s theorem and Böttcher’s chaotic maps as earlycontributions to the area

Lucjan Emil Böttcher (1872-1937): his life, work and academicstruggles

Paris 1918 or Leipzig and Lwów 1898?Böttcher’s role as one of the founders of holomorphic dynamics

Outline

What is holomorphic dynamics?

Holomorphic dynamics (in one variable) is an area of mathematicsstudying iterations of holomorphic maps on the Riemann sphere orcomplex affine plane. It was systematically developed by Frenchmathematicians Pierre Fatou and Gaston Julia, starting around 1918.Among its objects of study are so-called Julia sets and the Mandelbrotset, whose pictures have become widely known not only tomathematical audience.

The Mandelbrot set

An example of a Julia set

P(z) = z2 − 0.81000006198 + 0.344999969006i

Another example of a Julia set

P(z) = z2 + 1/4

Yet another example of a Julia set

Böttcher’s theorem

One of methods for creating pictures of Julia sets for polynomials useslevels (visible in the presented examples) of the modulus of so-calledBöttcher’s coordinate. Its existence follows from a theorem by LucjanEmil Böttcher, a Polish mathematician.

Theorem(Böttcher, 1898; 1904): Letf (z) = amzm + am+1zm+1 + ..., m > 2, am 6= 0, be an analyticfunction in a neighborhood of 0. Then there exists a conformal map Fof a neighborhood of 0 onto the unit disk, F (z) = z + bz2 + ...,satisfying the equation Ff (z) = [F (z)]m.

Böttcher’s theorem

One of methods for creating pictures of Julia sets for polynomials useslevels (visible in the presented examples) of the modulus of so-calledBöttcher’s coordinate. Its existence follows from a theorem by LucjanEmil Böttcher, a Polish mathematician.

Theorem(Böttcher, 1898; 1904): Letf (z) = amzm + am+1zm+1 + ..., m > 2, am 6= 0, be an analyticfunction in a neighborhood of 0. Then there exists a conformal map Fof a neighborhood of 0 onto the unit disk, F (z) = z + bz2 + ...,satisfying the equation Ff (z) = [F (z)]m.

Other contributions by Böttcher

Böttcher’s theorem is well known to specialists in holomorphicdynamics and in functional equations. It has many applications andgeneralizations. Another contribution by Böttcher are examples ofeverywhere chaotic rational maps (the Julia set is the whole sphere!)

But in fact there are many more results in holomorphic dynamics to befound in Böttcher’s work. In this talk we will discuss these results, theirperception by Böttcher’s contemporaries and their later developmentby other mathematicians.

Other contributions by Böttcher

Böttcher’s theorem is well known to specialists in holomorphicdynamics and in functional equations. It has many applications andgeneralizations. Another contribution by Böttcher are examples ofeverywhere chaotic rational maps (the Julia set is the whole sphere!)But in fact there are many more results in holomorphic dynamics to befound in Böttcher’s work. In this talk we will discuss these results, theirperception by Böttcher’s contemporaries and their later developmentby other mathematicians.

Highlights of Lucjan Emil Böttcher’s biography (1)

Born on January 7 (21), 1872 in Warsaw, in anEvangelical-Lutheran family.

Attends private real schools in Warsaw.

Passes maturity exam in the classical gymnasium in Łomza in1893. Enrolls in the Division of Mathematics and Physics of theImperial University of Warsaw. Attends lectures in mathematics,astronomy, physics and chemistry.

Expelled from the university in 1894 for participating in a Polishpatriotic manifestation.

Born on January 7 (21), 1872 in Warsaw, in anEvangelical-Lutheran family.

Attends private real schools in Warsaw.

Passes maturity exam in the classical gymnasium in Łomza in1893. Enrolls in the Division of Mathematics and Physics of theImperial University of Warsaw. Attends lectures in mathematics,astronomy, physics and chemistry.

Expelled from the university in 1894 for participating in a Polishpatriotic manifestation.

The seal of Imperial University at Warsaw

The statue of Col. Jan Kilinski in Warsaw

Moves to Lwów. Enrolls in the Division of Machine Construction inthe Lwów Polytechnic School. Passes the state exam in 1896.

Moves to Leipzig in 1897 to complete a course of studies inmathematics. Enrolls at the University of Leipzig and attendslectures in mathematics, physics and psychology.

Presents the dissertation “Beiträge zu der Theorie derIterationsrechnung", passes examinations and obtains the degreeof doctor of philosophy in 1898 (under the direction of Sophus Lie,one of the most important mathematicians of 19th century).

The seal of the University of Leipzig

Böttcher’s matriculation card

Böttcher’s PhD exam report with S. Lie’s signature

Böttcher’s dissertation

It took Böttcher 3 semesters to complete the course of study in Leipzigand prepare his dissertation. Influenced by Lie and his theories, he setout to study general iterations of maps in the framework of Lie groups.He chose the topic himself. Chapter I of Böttcher’s dissertationcontains some formal results in the intended direction. Chapter II isdevoted to iteration of rational functions over the Riemann sphere andcontains results and ideas which can be regarded as foundations ofholomorphic dynamics. Most results are stated without proofs.

Perception of Böttcher in Leipzig

The diversion from the main topic of dissertation and inclusion ofstatements without proofs were considered a flaw. Wilhelm Scheibnerrefused to submit a report on Böttcher’s dissertation. At the request ofLie (after correspondence with the university’s authorities) AdolphMayer served as another examiner. Lie’s opinion was as follows:

“At present I cannot recognize that the author has definitely managedto substantiate significant new results. Despite all of this, hisconsiderations, which testify to diligence and talent, have their value.(...)“Under the conditions mentioned above, I support the acceptance ofthe dissertation with evaluation II and admission to the oral exam".Böttcher’s final grade was “magna cum laude".

The diversion from the main topic of dissertation and inclusion ofstatements without proofs were considered a flaw. Wilhelm Scheibnerrefused to submit a report on Böttcher’s dissertation. At the request ofLie (after correspondence with the university’s authorities) AdolphMayer served as another examiner. Lie’s opinion was as follows:“At present I cannot recognize that the author has definitely managedto substantiate significant new results. Despite all of this, hisconsiderations, which testify to diligence and talent, have their value.(...)“Under the conditions mentioned above, I support the acceptance ofthe dissertation with evaluation II and admission to the oral exam".

Böttcher’s final grade was “magna cum laude".

The diversion from the main topic of dissertation and inclusion ofstatements without proofs were considered a flaw. Wilhelm Scheibnerrefused to submit a report on Böttcher’s dissertation. At the request ofLie (after correspondence with the university’s authorities) AdolphMayer served as another examiner. Lie’s opinion was as follows:“At present I cannot recognize that the author has definitely managedto substantiate significant new results. Despite all of this, hisconsiderations, which testify to diligence and talent, have their value.(...)“Under the conditions mentioned above, I support the acceptance ofthe dissertation with evaluation II and admission to the oral exam".Böttcher’s final grade was “magna cum laude".

Returns to Lwów and takes a position of an assistant in the LwówPolytechnic School in 1901. Submits a request for habilitation atthe Lwów University, which is denied.

Becomes an adjunct in the Lwów Polytechnic School in 1910 andobtains veniam legendi in mathematics in 1911. Makes a requestthat this licence be also recognized at the Lwów University,without success.

Lectures on mathematics for engineers and on theoreticalmechanics. Takes part in activities of scientific societies.Publishes articles in mathematics (total known number 19),mathematical education, logic and mechanics, as well as lecturenotes, popularization pieces and high school textbooks. Makestwo more attempts to obtain habilitation, both unsuccessful.

Retires from the Lwów Polytechnic School in 1935. Dies in Lwówon May 29, 1937.

Lviv (Lwów/Lvov) Polytechnics

Böttcher’s registry card at Lwów Polytechnics

Böttcher’s application for habilitation

Committee’s signatures

Böttcher’s first attempt at habilitation

The first attempt Böttcher made to obtain habilitation at the LwówUniversity was accompanied by two publications: "Principles ofiterational calculus, part three" (Prace Matematyczno-Fizyczne, v. XII(1901), p. 95-111) and "On properties of some functionaldeterminants" (Rozprawy Wydziału Matematyczno-PrzyrodniczegoAkademii Umiejetnosci w Krakowie, v. 38 (general volume) (1901);series II, v. 18(1901), 382-389).

The second paper was not related to holomorphic dynamics; the firstone was, but did not contain original results.The committee, whose members were Józef Puzyna, Jan Rajewski,Marian Smoluchowski and the dean Ludwik Finkel, deemed the resultscorrect but insufficient, and the decision, made on February 6, 1902,was not to admit Böttcher to habilitation at that time but rather wait formore results from him.

The first attempt Böttcher made to obtain habilitation at the LwówUniversity was accompanied by two publications: "Principles ofiterational calculus, part three" (Prace Matematyczno-Fizyczne, v. XII(1901), p. 95-111) and "On properties of some functionaldeterminants" (Rozprawy Wydziału Matematyczno-PrzyrodniczegoAkademii Umiejetnosci w Krakowie, v. 38 (general volume) (1901);series II, v. 18(1901), 382-389).The second paper was not related to holomorphic dynamics; the firstone was, but did not contain original results.

The committee, whose members were Józef Puzyna, Jan Rajewski,Marian Smoluchowski and the dean Ludwik Finkel, deemed the resultscorrect but insufficient, and the decision, made on February 6, 1902,was not to admit Böttcher to habilitation at that time but rather wait formore results from him.

The first attempt Böttcher made to obtain habilitation at the LwówUniversity was accompanied by two publications: "Principles ofiterational calculus, part three" (Prace Matematyczno-Fizyczne, v. XII(1901), p. 95-111) and "On properties of some functionaldeterminants" (Rozprawy Wydziału Matematyczno-PrzyrodniczegoAkademii Umiejetnosci w Krakowie, v. 38 (general volume) (1901);series II, v. 18(1901), 382-389).The second paper was not related to holomorphic dynamics; the firstone was, but did not contain original results.The committee, whose members were Józef Puzyna, Jan Rajewski,Marian Smoluchowski and the dean Ludwik Finkel, deemed the resultscorrect but insufficient, and the decision, made on February 6, 1902,was not to admit Böttcher to habilitation at that time but rather wait formore results from him.

More struggles for habilitation

In 1918 Böttcher submitted 6 publications with his request forhabilitation, among them Glavn"ishiye zakony skhodimosti iteratsii i ikhprilozheniya k" analizu [The principal laws of convergence of iteratesand their application to analysis], Bulletin de la Societe Physico-Mathematique de Kasan, tome XIII (1, 1903), p.137, XIV (2, 1904), p.155-200, XIV (3, 1904), p. 201-234. This paper partially overlaps withBöttchers dissertation and with another major paper, published inPolish in 1899. It contains Böttcher’s theorem and became widely citedafter Joseph Fels Ritt referred to it in 1921.The committee found many shortcomings and errors in the submittedworks.

This attempt at habilitation (and the next and last one, in 1919) alsofailed.

More struggles for habilitation

In 1918 Böttcher submitted 6 publications with his request forhabilitation, among them Glavn"ishiye zakony skhodimosti iteratsii i ikhprilozheniya k" analizu [The principal laws of convergence of iteratesand their application to analysis], Bulletin de la Societe Physico-Mathematique de Kasan, tome XIII (1, 1903), p.137, XIV (2, 1904), p.155-200, XIV (3, 1904), p. 201-234. This paper partially overlaps withBöttchers dissertation and with another major paper, published inPolish in 1899. It contains Böttcher’s theorem and became widely citedafter Joseph Fels Ritt referred to it in 1921.The committee found many shortcomings and errors in the submittedworks.This attempt at habilitation (and the next and last one, in 1919) alsofailed.

Negative perception of Böttcher’s results (1)

Here are excerpts of the habilitation committee’s opinion:

“Despite great verve and determination, Dr. Böttcher’s works do notyield any positive scientific results. There are many formalmanipulations and computations in them; essential difficulties areusually dismissed with a few words without deeper treatment. Thecontent and character diverges significantly from modern research."

Here are excerpts of the habilitation committee’s opinion:“Despite great verve and determination, Dr. Böttcher’s works do notyield any positive scientific results. There are many formalmanipulations and computations in them; essential difficulties areusually dismissed with a few words without deeper treatment. Thecontent and character diverges significantly from modern research."

“The method used by the Candidate in his works cannot be consideredscientific. The author works with undefined, or ill-defined, notions (e.g.,the notion of an iteration with an arbitrary exponent), and the majorityof the results he achieves are transformations of one problem intoanother, no less difficult. In the proofs there are moreover illegitimateconclusions, or even fundamental mistakes."(...)

In principle, the committee was right: Böttcher only sketched theproofs of his deeper results, and often did not offer any justification forhis conclusions.

“The method used by the Candidate in his works cannot be consideredscientific. The author works with undefined, or ill-defined, notions (e.g.,the notion of an iteration with an arbitrary exponent), and the majorityof the results he achieves are transformations of one problem intoanother, no less difficult. In the proofs there are moreover illegitimateconclusions, or even fundamental mistakes."(...)In principle, the committee was right: Böttcher only sketched theproofs of his deeper results, and often did not offer any justification forhis conclusions.

What the committee did not see

It is hard to determine whether Böttcher’s paper published in Russiancould be understood by the committee members. There were also twoearlier publications related to holomorphic dynamics, which were notsubmitted with any of Böttcher’s applications for habilitation:

(1) Beiträge zu der Theorie der Iterationsrechnung, published byOswald Schmidt, Leipzig, pp.78, 1898 (doctoral dissertation);(2) Zasady rachunku iteracyjnego (czesc pierwsza i czesc druga)[Principles of iterational calculus (part one and two)], PraceMatematyczno-Fizyczne, vol. X (1899,1900), pp. 65-86, 86-101.These works contain many fundamental notions and partial results ofholomorphic dynamics, later re-developed independently by Fatou,Julia, Lattés and Pincherle.

It is hard to determine whether Böttcher’s paper published in Russiancould be understood by the committee members. There were also twoearlier publications related to holomorphic dynamics, which were notsubmitted with any of Böttcher’s applications for habilitation:(1) Beiträge zu der Theorie der Iterationsrechnung, published byOswald Schmidt, Leipzig, pp.78, 1898 (doctoral dissertation);

(2) Zasady rachunku iteracyjnego (czesc pierwsza i czesc druga)[Principles of iterational calculus (part one and two)], PraceMatematyczno-Fizyczne, vol. X (1899,1900), pp. 65-86, 86-101.These works contain many fundamental notions and partial results ofholomorphic dynamics, later re-developed independently by Fatou,Julia, Lattés and Pincherle.

It is hard to determine whether Böttcher’s paper published in Russiancould be understood by the committee members. There were also twoearlier publications related to holomorphic dynamics, which were notsubmitted with any of Böttcher’s applications for habilitation:(1) Beiträge zu der Theorie der Iterationsrechnung, published byOswald Schmidt, Leipzig, pp.78, 1898 (doctoral dissertation);(2) Zasady rachunku iteracyjnego (czesc pierwsza i czesc druga)[Principles of iterational calculus (part one and two)], PraceMatematyczno-Fizyczne, vol. X (1899,1900), pp. 65-86, 86-101.

These works contain many fundamental notions and partial results ofholomorphic dynamics, later re-developed independently by Fatou,Julia, Lattés and Pincherle.

It is hard to determine whether Böttcher’s paper published in Russiancould be understood by the committee members. There were also twoearlier publications related to holomorphic dynamics, which were notsubmitted with any of Böttcher’s applications for habilitation:(1) Beiträge zu der Theorie der Iterationsrechnung, published byOswald Schmidt, Leipzig, pp.78, 1898 (doctoral dissertation);(2) Zasady rachunku iteracyjnego (czesc pierwsza i czesc druga)[Principles of iterational calculus (part one and two)], PraceMatematyczno-Fizyczne, vol. X (1899,1900), pp. 65-86, 86-101.These works contain many fundamental notions and partial results ofholomorphic dynamics, later re-developed independently by Fatou,Julia, Lattés and Pincherle.

Böttcher’s dissertation

Böttcher’s doctoral dissertation introduces:-the study of individual orbits of (iterated) rational maps, of theirconvergence and the limits that occur;-the study of “regions of convergence" (later called Fatou components)and their boundaries (Julia sets); method of determining theboundaries using backward iteration;-an example of an everywhere chaotic map, i.e., a map without regionsof convergence, constructed by means of elliptic functions (a similarexample was given in 1918 by Samuel Lattés);-some observations about preperiodic points (nowadays important inthe study of parameter spaces and in arithmetic dynamics).

“Zasady rachunku iteracyjnego", cz. I i cz. II

This paper contains:-a different example of an everywhere chaotic map and a sketch ofproof of its chaotic behavior;-examples of Julia sets (called “boundary curves") for monomials andChebyshev polynomials; study of their simple dynamical properties ,e.g„ density of periodic points;-mention of irrationally neutral periodic points;-formulation of an exact upper bound for the number of (periodic)“regions of convergence" of a rational map in terms of the number ofits critical points (now known as Fatou-Shishikura inequality);-the first formulation of Böttcher’s theorem and a sketch of its proof.

Böttcher as a founder of holomorphic dynamics

All these topics re-emerged after 1918, when Böttcher’s early workwas all but forgotten. Pierre Fatou and Gaston Julia (independently)developed similar notions and facts using the framework of normalfamilies, taking advantage of the theory formulated by Paul Montel.Their work started the systematic development of holomorphicdynamics as a new discipline of mathematics. Nowadays it is still avery active area of research. Böttcher pioneered many of itsfundamental ideas and results, so despite some flaws in his works heshould be regarded as one of the founders of this discipline.

Acknowledgments

This talk has its origin in a joint project with Stanisław Domoradzkiconcerning Lucjan Emil Böttcher and his mathematical legacy.Documents related to Böttcher presented here come from theUniversity Archive in Leipzig and Lviv District Archive (found by S.Domoradzki).Pictures from Google images and Wikimedia Commons.A preliminary written version of this talk can be found here:http://arxiv.org/pdf/1307.7778.pdf

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