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 early
contributions to the area

Outline
Lucjan Emil Böttcher (1872-1937): his life, work and academic
struggles

Outline
Lucjan Emil Böttcher (1872-1937): his life, work and academic
struggles
Paris 1918 or Leipzig and Lwów 1898?
Böttcher’s role as one of the founders of holomorphic dynamics

What is holomorphic dynamics?
Holomorphic dynamics (in one variable) is an area of mathematics
studying iterations of holomorphic maps on the Riemann sphere or
complex afﬁne plane. It was systematically developed by French
mathematicians Pierre Fatou and Gaston Julia, starting around 1918.
Among its objects of study are so-called Julia sets and the Mandelbrot
set, whose pictures have become widely known not only to
mathematical 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 uses
levels (visible in the presented examples) of the modulus of so-called
Böttcher’s coordinate. Its existence follows from a theorem by Lucjan
Emil Böttcher, a Polish mathematician.

Böttcher’s theorem
One of methods for creating pictures of Julia sets for polynomials uses
levels (visible in the presented examples) of the modulus of so-called
Böttcher’s coordinate. Its existence follows from a theorem by Lucjan
Emil Böttcher, a Polish mathematician.
Theorem
(Böttcher, 1898; 1904): Let
f(z) = amzm + am+1zm+1 + ..., m 2, am = 0, be an analytic
function in a neighborhood of 0. Then there exists a conformal map F
of 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 holomorphic
dynamics and in functional equations. It has many applications and
generalizations. Another contribution by Böttcher are examples of
everywhere chaotic rational maps (the Julia set is the whole sphere!)

Other contributions by Böttcher
Böttcher’s theorem is well known to specialists in holomorphic
dynamics and in functional equations. It has many applications and
generalizations. Another contribution by Böttcher are examples of
everywhere chaotic rational maps (the Julia set is the whole sphere!)
But in fact there are many more results in holomorphic dynamics to be
found in Böttcher’s work. In this talk we will discuss these results, their
perception by Böttcher’s contemporaries and their later development
by other mathematicians.

Highlights of Lucjan Emil Böttcher’s biography (1)
Born on January 7 (21), 1872 in Warsaw, in an
Evangelical-Lutheran family.
Attends private real schools in Warsaw.
Passes maturity exam in the classical gymnasium in Łom˙za in
1893. Enrolls in the Division of Mathematics and Physics of the
Imperial University of Warsaw. Attends lectures in mathematics,
astronomy, physics and chemistry.

Born on January 7 (21), 1872 in Warsaw, in an
Evangelical-Lutheran family.
Attends private real schools in Warsaw.
Passes maturity exam in the classical gymnasium in Łom˙za in
1893. Enrolls in the Division of Mathematics and Physics of the
Imperial University of Warsaw. Attends lectures in mathematics,
astronomy, physics and chemistry.
Expelled from the university in 1894 for participating in a Polish
patriotic manifestation.

The seal of Imperial University at Warsaw

The statue of Col. Jan Kili´nski in Warsaw

Moves to Lwów. Enrolls in the Division of Machine Construction in
the Lwów Polytechnic School. Passes the state exam in 1896.
Moves to Leipzig in 1897 to complete a course of studies in
mathematics. Enrolls at the University of Leipzig and attends
lectures in mathematics, physics and psychology.
Presents the dissertation “Beiträge zu der Theorie der
Iterationsrechnung", passes examinations and obtains the degree
of 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 Leipzig
and prepare his dissertation. Inﬂuenced by Lie and his theories, he set
out to study general iterations of maps in the framework of Lie groups.
He chose the topic himself. Chapter I of Böttcher’s dissertation
contains some formal results in the intended direction. Chapter II is
devoted to iteration of rational functions over the Riemann sphere and
contains results and ideas which can be regarded as foundations of
holomorphic dynamics. Most results are stated without proofs.

Perception of Böttcher in Leipzig
The diversion from the main topic of dissertation and inclusion of
statements without proofs were considered a ﬂaw. Wilhelm Scheibner
refused to submit a report on Böttcher’s dissertation. At the request of
Lie (after correspondence with the university’s authorities) Adolph
Mayer served as another examiner. Lie’s opinion was as follows:

“At present I cannot recognize that the author has deﬁnitely managed
to substantiate signiﬁcant new results. Despite all of this, his
considerations, which testify to diligence and talent, have their value.
(...)
“Under the conditions mentioned above, I support the acceptance of
the dissertation with evaluation II and admission to the oral exam".

“At present I cannot recognize that the author has definitely managed
to substantiate significant new results. Despite all of this, his
considerations, which testify to diligence and talent, have their value.
(...)
“Under the conditions mentioned above, I support the acceptance of
the 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 Lwów
Polytechnic School in 1901. Submits a request for habilitation at
the Lwów University, which is denied.

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

without success.
Lectures on mathematics for engineers and on theoretical
mechanics. Takes part in activities of scientiﬁc societies.
Publishes articles in mathematics (total known number 19),
mathematical education, logic and mechanics, as well as lecture
notes, popularization pieces and high school textbooks. Makes
two more attempts to obtain habilitation, both unsuccessful.

without success.
Lectures on mathematics for engineers and on theoretical
mechanics. Takes part in activities of scientiﬁc societies.
Publishes articles in mathematics (total known number 19),
mathematical education, logic and mechanics, as well as lecture
notes, popularization pieces and high school textbooks. Makes
two more attempts to obtain habilitation, both unsuccessful.
Retires from the Lwów Polytechnic School in 1935. Dies in Lwów
on 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 Lwów
University was accompanied by two publications: "Principles of
iterational calculus, part three" (Prace Matematyczno-Fizyczne, v. XII
(1901), p. 95-111) and "On properties of some functional
determinants" (Rozprawy Wydziału Matematyczno-Przyrodniczego
Akademii Umieje¸tno´sci w Krakowie, v. 38 (general volume) (1901);
series II, v. 18(1901), 382-389).

series II, v. 18(1901), 382-389).
The second paper was not related to holomorphic dynamics; the ﬁrst
one was, but did not contain original results.

series II, v. 18(1901), 382-389).
The second paper was not related to holomorphic dynamics; the ﬁrst
one 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 results
correct but insufﬁcient, and the decision, made on February 6, 1902,
was not to admit Böttcher to habilitation at that time but rather wait for
more results from him.

More struggles for habilitation
In 1918 Böttcher submitted 6 publications with his request for
habilitation, among them Glavn"ishiye zakony skhodimosti iteratsii i ikh
prilozheniya k" analizu [The principal laws of convergence of iterates
and 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 with
Böttchers dissertation and with another major paper, published in
Polish in 1899. It contains Böttcher’s theorem and became widely cited
after Joseph Fels Ritt referred to it in 1921.
The committee found many shortcomings and errors in the submitted
works.

More struggles for habilitation
In 1918 Böttcher submitted 6 publications with his request for
habilitation, among them Glavn"ishiye zakony skhodimosti iteratsii i ikh
prilozheniya k" analizu [The principal laws of convergence of iterates
and 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 with
Böttchers dissertation and with another major paper, published in
Polish in 1899. It contains Böttcher’s theorem and became widely cited
after Joseph Fels Ritt referred to it in 1921.
The committee found many shortcomings and errors in the submitted
works.
This attempt at habilitation (and the next and last one, in 1919) also
failed.

Negative perception of Böttcher’s results (1)
Here are excerpts of the habilitation committee’s opinion:

Here are excerpts of the habilitation committee’s opinion:
“Despite great verve and determination, Dr. Böttcher’s works do not
yield any positive scientific results. There are many formal
manipulations and computations in them; essential difficulties are
usually dismissed with a few words without deeper treatment. The
content and character diverges significantly from modern research."

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

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

What the committee did not see
It is hard to determine whether Böttcher’s paper published in Russian
could be understood by the committee members. There were also two
earlier publications related to holomorphic dynamics, which were not
submitted with any of Böttcher’s applications for habilitation:

(1) Beiträge zu der Theorie der Iterationsrechnung, published by
Oswald Schmidt, Leipzig, pp.78, 1898 (doctoral dissertation);

(2) Zasady rachunku iteracyjnego (cze¸ ´s´c pierwsza i cze¸ ´s´c druga)
[Principles of iterational calculus (part one and two)], Prace
Matematyczno-Fizyczne, vol. X (1899,1900), pp. 65-86, 86-101.

(2) Zasady rachunku iteracyjnego (cze¸ ´s´c pierwsza i cze¸ ´s´c druga)
[Principles of iterational calculus (part one and two)], Prace
Matematyczno-Fizyczne, vol. X (1899,1900), pp. 65-86, 86-101.
These works contain many fundamental notions and partial results of
holomorphic 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 their
convergence and the limits that occur;
-the study of “regions of convergence" (later called Fatou components)
and their boundaries (Julia sets); method of determining the
boundaries using backward iteration;
-an example of an everywhere chaotic map, i.e., a map without regions
of convergence, constructed by means of elliptic functions (a similar
example was given in 1918 by Samuel Lattés);
-some observations about preperiodic points (nowadays important in
the 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 of
proof of its chaotic behavior;
-examples of Julia sets (called “boundary curves") for monomials and
Chebyshev 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 of
its 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 work
was all but forgotten. Pierre Fatou and Gaston Julia (independently)
developed similar notions and facts using the framework of normal
families, taking advantage of the theory formulated by Paul Montel.
Their work started the systematic development of holomorphic
dynamics as a new discipline of mathematics. Nowadays it is still a
very active area of research. Böttcher pioneered many of its
fundamental ideas and results, so despite some ﬂaws in his works he
should be regarded as one of the founders of this discipline.

Acknowledgments
This talk has its origin in a joint project with Stanisław Domoradzki
concerning Lucjan Emil Böttcher and his mathematical legacy.
Documents related to Böttcher presented here come from the
University 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

Lucjan Emil Boettcher- The Polish pioneer of holomorphic dynamics

Recommandé

Recommandé

Contenu connexe

Similaire à Lucjan Emil Boettcher- The Polish pioneer of holomorphic dynamics

Similaire à Lucjan Emil Boettcher- The Polish pioneer of holomorphic dynamics (9)

Dernier

Dernier (20)

Lucjan Emil Boettcher- The Polish pioneer of holomorphic dynamics