Lucjan Emil Böttcher (1872-1937):
the Polish pioneer of holomorphic dynamics
Małgorzata Stawiska-Friedland
Mathematical Re...
Outline
Holomorphic dynamics;
Böttcher’s theorem and Böttcher’s chaotic maps as early
contributions to the area
M. Stawisk...
Outline
Holomorphic dynamics;
Böttcher’s theorem and Böttcher’s chaotic maps as early
contributions to the area
Lucjan Emi...
Outline
Holomorphic dynamics;
Böttcher’s theorem and Böttcher’s chaotic maps as early
contributions to the area
Lucjan Emi...
What is holomorphic dynamics?
Holomorphic dynamics (in one variable) is an area of mathematics
studying iterations of holo...
The Mandelbrot set
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 4 / 32
An example of a Julia set
P(z) = z2 − 0.81000006198 + 0.344999969006i
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Bö...
Another example of a Julia set
P(z) = z2 + 1/4
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 6 / ...
Yet another example of a Julia set
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 7 / 32
Böttcher’s theorem
One of methods for creating pictures of Julia sets for polynomials uses
levels (visible in the presente...
Böttcher’s theorem
One of methods for creating pictures of Julia sets for polynomials uses
levels (visible in the presente...
Other contributions by Böttcher
Böttcher’s theorem is well known to specialists in holomorphic
dynamics and in functional ...
Other contributions by Böttcher
Böttcher’s theorem is well known to specialists in holomorphic
dynamics and in functional ...
Highlights of Lucjan Emil Böttcher’s biography (1)
Born on January 7 (21), 1872 in Warsaw, in an
Evangelical-Lutheran fami...
Highlights of Lucjan Emil Böttcher’s biography (1)
Born on January 7 (21), 1872 in Warsaw, in an
Evangelical-Lutheran fami...
The seal of Imperial University at Warsaw
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 11 / 32
The statue of Col. Jan Kili´nski in Warsaw
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 12 / 32
Highlights of Lucjan Emil Böttcher’s biography (2)
Moves to Lwów. Enrolls in the Division of Machine Construction in
the L...
The seal of the University of Leipzig
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 14 / 32
Böttcher’s matriculation card
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 15 / 32
Böttcher’s PhD exam report with S. Lie’s signature
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 ...
Böttcher’s dissertation
It took Böttcher 3 semesters to complete the course of study in Leipzig
and prepare his dissertati...
Perception of Böttcher in Leipzig
The diversion from the main topic of dissertation and inclusion of
statements without pr...
Perception of Böttcher in Leipzig
The diversion from the main topic of dissertation and inclusion of
statements without pr...
Perception of Böttcher in Leipzig
The diversion from the main topic of dissertation and inclusion of
statements without pr...
Highlights of Lucjan Emil Böttcher’s biography (3)
Returns to Lwów and takes a position of an assistant in the Lwów
Polyte...
Highlights of Lucjan Emil Böttcher’s biography (3)
Returns to Lwów and takes a position of an assistant in the Lwów
Polyte...
Highlights of Lucjan Emil Böttcher’s biography (3)
Returns to Lwów and takes a position of an assistant in the Lwów
Polyte...
Highlights of Lucjan Emil Böttcher’s biography (3)
Returns to Lwów and takes a position of an assistant in the Lwów
Polyte...
Lviv (Lwów/Lvov) Polytechnics
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 20 / 32
Böttcher’s registry card at Lwów Polytechnics
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 21 / ...
Böttcher’s application for habilitation
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 22 / 32
Committee’s signatures
M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 23 / 32
Böttcher’s first attempt at habilitation
The first attempt Böttcher made to obtain habilitation at the Lwów
University was a...
Böttcher’s first attempt at habilitation
The first attempt Böttcher made to obtain habilitation at the Lwów
University was a...
Böttcher’s first attempt at habilitation
The first attempt Böttcher made to obtain habilitation at the Lwów
University was a...
More struggles for habilitation
In 1918 Böttcher submitted 6 publications with his request for
habilitation, among them Gl...
More struggles for habilitation
In 1918 Böttcher submitted 6 publications with his request for
habilitation, among them Gl...
Negative perception of Böttcher’s results (1)
Here are excerpts of the habilitation committee’s opinion:
M. Stawiska-Fried...
Negative perception of Böttcher’s results (1)
Here are excerpts of the habilitation committee’s opinion:
“Despite great ve...
Negative perception of Böttcher’s results (2)
“The method used by the Candidate in his works cannot be considered
scientifi...
Negative perception of Böttcher’s results (2)
“The method used by the Candidate in his works cannot be considered
scientifi...
What the committee did not see
It is hard to determine whether Böttcher’s paper published in Russian
could be understood b...
What the committee did not see
It is hard to determine whether Böttcher’s paper published in Russian
could be understood b...
What the committee did not see
It is hard to determine whether Böttcher’s paper published in Russian
could be understood b...
What the committee did not see
It is hard to determine whether Böttcher’s paper published in Russian
could be understood b...
Böttcher’s dissertation
Böttcher’s doctoral dissertation introduces:
-the study of individual orbits of (iterated) rationa...
“Zasady rachunku iteracyjnego", cz. I i cz. II
This paper contains:
-a different example of an everywhere chaotic map and ...
Böttcher as a founder of holomorphic dynamics
All these topics re-emerged after 1918, when Böttcher’s early work
was all b...
Acknowledgments
This talk has its origin in a joint project with Stanisław Domoradzki
concerning Lucjan Emil Böttcher and ...
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Lucjan Emil Boettcher- The Polish pioneer of holomorphic dynamics

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Lucjan Emil Boettcher- The Polish pioneer of holomorphic dynamics

  1. 1. 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
  2. 2. Outline Holomorphic dynamics; Böttcher’s theorem and Böttcher’s chaotic maps as early contributions to the area M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 2 / 32
  3. 3. Outline Holomorphic dynamics; Böttcher’s theorem and Böttcher’s chaotic maps as early contributions to the area Lucjan Emil Böttcher (1872-1937): his life, work and academic struggles M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 2 / 32
  4. 4. Outline Holomorphic dynamics; Böttcher’s theorem and Böttcher’s chaotic maps as early contributions to the area 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 M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 2 / 32
  5. 5. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 3 / 32
  6. 6. The Mandelbrot set M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 4 / 32
  7. 7. An example of a Julia set P(z) = z2 − 0.81000006198 + 0.344999969006i M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 5 / 32
  8. 8. Another example of a Julia set P(z) = z2 + 1/4 M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 6 / 32
  9. 9. Yet another example of a Julia set M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 7 / 32
  10. 10. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 8 / 32
  11. 11. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 8 / 32
  12. 12. 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!) M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 9 / 32
  13. 13. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 9 / 32
  14. 14. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 10 / 32
  15. 15. 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. Expelled from the university in 1894 for participating in a Polish patriotic manifestation. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 10 / 32
  16. 16. The seal of Imperial University at Warsaw M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 11 / 32
  17. 17. The statue of Col. Jan Kili´nski in Warsaw M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 12 / 32
  18. 18. Highlights of Lucjan Emil Böttcher’s biography (2) 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). M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 13 / 32
  19. 19. The seal of the University of Leipzig M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 14 / 32
  20. 20. Böttcher’s matriculation card M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 15 / 32
  21. 21. Böttcher’s PhD exam report with S. Lie’s signature M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 16 / 32
  22. 22. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 17 / 32
  23. 23. 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: M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 18 / 32
  24. 24. 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". M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 18 / 32
  25. 25. 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". Böttcher’s final grade was “magna cum laude". M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 18 / 32
  26. 26. Highlights of Lucjan Emil Böttcher’s biography (3) 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 19 / 32
  27. 27. Highlights of Lucjan Emil Böttcher’s biography (3) 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 19 / 32
  28. 28. Highlights of Lucjan Emil Böttcher’s biography (3) 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. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 19 / 32
  29. 29. Highlights of Lucjan Emil Böttcher’s biography (3) 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. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 19 / 32
  30. 30. Lviv (Lwów/Lvov) Polytechnics M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 20 / 32
  31. 31. Böttcher’s registry card at Lwów Polytechnics M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 21 / 32
  32. 32. Böttcher’s application for habilitation M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 22 / 32
  33. 33. Committee’s signatures M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 23 / 32
  34. 34. 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). M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 24 / 32
  35. 35. 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). The second paper was not related to holomorphic dynamics; the first one was, but did not contain original results. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 24 / 32
  36. 36. 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). 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 24 / 32
  37. 37. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 25 / 32
  38. 38. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 25 / 32
  39. 39. Negative perception of Böttcher’s results (1) Here are excerpts of the habilitation committee’s opinion: M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 26 / 32
  40. 40. 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 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." M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 26 / 32
  41. 41. Negative perception of Böttcher’s results (2) “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."(...) M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 27 / 32
  42. 42. Negative perception of Böttcher’s results (2) “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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 27 / 32
  43. 43. 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: M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 28 / 32
  44. 44. 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); M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 28 / 32
  45. 45. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 28 / 32
  46. 46. 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. These works contain many fundamental notions and partial results of holomorphic dynamics, later re-developed independently by Fatou, Julia, Lattés and Pincherle. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 28 / 32
  47. 47. 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). M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 29 / 32
  48. 48. “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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 30 / 32
  49. 49. 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. M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 31 / 32
  50. 50. 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 M. Stawiska-Friedland (Math. Reviews) Lucjan Emil Böttcher Kraków 2013 32 / 32

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