The theory of complex dynamics, whose roots lie in 19th-century
studies of the iteration of complex function conducted by Koenigs,
Schoder, 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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