1. INTRODUCTIO N

FIGURE

1.3. Sensitivit y t o initia l condition s

FIGURE

1.4. Topologica l transitivit y

FIGURE

1.5. Nonwanderin g point s

Another concep t simila r t o chaoti c i s ergodicity. Suppos e F i s a measurabl e

self-map o f a compac t topologica l spac e K. A Bore l measur e / x on K i s sai d t o

be invarian t i f

/

JL(F'~1(E))

= fi(E) fo r al l Bore l set s E C K. Suppos e tha t / x is a n

invariant probabilit y measur e o n K. W e say tha t / x is ergodic i f wheneve r E i s a n

invariant Bore l set , i.e . F(E) = F-

X(E)

= E, the n fi(E) = 0 or 1.

Fixed point s pla y a specia l rol e i n dynamics . Root s p o f R = 0 ar e fixed

points fo r th e ma p F fro m Newton' s method . Suppos e tha t F i s a holomorphi c

self-map o f a comple x manifol d M. Suppos e tha t p £ M i s a fixed poin t o f F.

If al l th e eigenvalue s o f F'{p) ar e strictl y les s tha n one , the n p i s a n attractin g

fixed poin t an d ther e i s an ope n set , th e attractin g basi n o f p, consistin g o f point s