Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906-1942

Portada
American Mathematical Soc., 2012 - 454 pàgines
The theory of complex dynamics, whose roots lie in 19th-century studies of the iteration of complex function conducted by Koenigs, Schroder, and others, flourished remarkably during the first half of the 20th century, when many of the central ideas and techniques of the subject developed. This book by Alexander, Iavernaro, and Rosa paints a robust picture of the field of complex dynamics between 1906 and 1942 through detailed discussions of the work of Fatou, Julia, Siegel, and several others. A recurrent theme of the authors' treatment is the center problem in complex dynamics. They present its complete history during this period and, in so doing, bring out analogies between complex dynamics and the study of differential equations, in particular, the problem of stability in Hamiltonian systems. Among these analogies are the use of iteration and problems involving small divisors which the authors examine in the work of Poincare and others, linking them to complex dynamics, principally via the work of Samuel Lattes, in the early 1900s, and Jurgen Moser, in the 1960s. Many details will be new to the reader, such as a history of Lattes functions (functions whose Julia set equals the Riemann sphere), complex dynamics in the United States around the time of World War I, a survey of complex dynamics around the world in the 1920s and 1930s, a discussion of the dynamical programs of Fatou and Julia during the 1920s, and biographical material on several key figures. The book contains graphical renderings of many of the mathematical objects the authors discuss, including some of the intriguing fractals Fatou and Julia studied, and concludes with several appendices by current researchers in complex dynamics which collectively attest to the impact of the work of Fatou, Julia, and others upon the present-day study.
 

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Continguts

A Complex Dynamics Primer
3
1
10
Dynamics of a Complex History
23
Iteration and Differential Equations
33
3
39
Iteration and Differential Equations
67
The United States
95
5
103
Fatou and Julia in the 1920s
213
The German Wave
229
Siegel the Center Problem and KAM Theory
253
Iteratin Around the Globe
267
Appendix A Report on the Grand Prix des Sciences Mathématiques
281
Singular Lines of Analytic Functions
291
Appendix E Curves of Julia
297
Appendix G The DenjoyWolff Theorem
303

The Road to
113
11
117
Works Written for the Grand Prix
131
20
141
29
150
33
158
41
164
49
172
Iteration in Italy
181
The Giants Fall
197
58
207
Koebe and Uniformization
313
Appendix K The Last 60 Years in Permutable Maps
325
Understanding Julia Sets of Entire Maps
331
A Biographical Sketch
341
A Biographical Sketch
351
59
355
Appendix O Selected Biographies
359
Appendix P Remarks on Computer Graphics
371
Glossary
379
Index
433
Copyright

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