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## Main Question or Discussion Point

D is be a bounded domain in the complex plane. Suppose f : D -->D is a holomorphic automorphism (conformal bijection). Now define f_n(z) = f(f(f(f ..(z) (composed n times ).

Trying (and failing) to show:

(i) the sequence {f_n} has a subsequence that converges either to a constant

or to an automorphism of D

also

(ii) If the whole sequence {f_n} converges to g, then f(z)= z identically. $f(z)\equiv z

Trying (and failing) to show:

(i) the sequence {f_n} has a subsequence that converges either to a constant

or to an automorphism of D

also

(ii) If the whole sequence {f_n} converges to g, then f(z)= z identically. $f(z)\equiv z