CHAPTER 1
Introduction
The purpos e o f thi s lectur e serie s wa s t o introduc e th e audienc e t o th e literatur e
on comple x dynamic s i n highe r dimension . Thi s CBM S lectur e serie s wa s hel d i n
Albany, New York, June 1994. Som e of the lectures were up-dated version s of earlier
lectures given in Montreal 1993, jointly wit h Nessi m Sibony ([FS8]) . Th e Montrea l
lectures were more based on pluripotential theory, while the present lectures avoided
pluripotential theor y completely . W e hope t o giv e a n alternativ e expansio n o f th e
Montreal lectures , basin g comple x dynamic s i n higher dimensio n systematicall y o n
pluripotential theory .
What w e ar e tryin g t o d o wit h thes e note s i s t o provid e a n eas y t o rea d in -
troduction t o th e field, a n introductio n whic h motivate s th e topics . Moreover , th e
monograph shoul d poin t th e reade r toward s th e technicall y mor e advance d litera -
ture. I t is my feeling that mathematician s ca n read arbitrarily complicate d materia l
once they ar e motivated .
We wil l star t ou r introductio n b y choosin g a basi c proble m whic h everybod y
has seen before. Th e investigation of this problem will lead us naturally to studyin g
Complex Dynamic s i n Higher Dimension . I n order t o understand th e proble m bet -
ter, we are naturally lea d to some dynamical questions . Thes e dynamical question s
are the one s which hav e been studie d i n the literature .
Our basi c proble m i s t o find root s o f equations . Thi s is not just a peda -
gogical tric k becaus e thi s wa s alread y a motivating proble m fo r comple x dynamic s
in the las t century . A main too l for finding root s of equations i s Newton's method .
The ide a o f Newton's metho d i s to gues s a root an d us e this gues s to find a bette r
guess.
The first on e to study comple x dynamics was Schroder, 1870, 1871. Althoug h
Newton's metho d can , i n it s simples t form , b e trace d bac k t o th e Babylonians , h e
was th e first t o appl y i t t o stud y comple x root s o f holomorphi c polynomial s o f
one complex variable. (Se e ([Scl] ) an d ([Sc2]). ) W e tak e a s ou r startin g poin t
the proble m t o describ e thos e initia l guesse s whic h lea d t o a root .
To stud y thi s problem , le t u s introduc e som e notation . W e ca n wor k i n an y
dimension, bu t fo r simplicity w e will mostly restric t ou r discussions to two complex
variables. Moreover , we will restrict ou r discussion to roots of polynomial equations.
1
http://dx.doi.org/10.1090/cbms/087/01
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