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Give an explicit description of all the biholomorphic self maps of $\Omega$, for:

  1. $\Omega = \mathbb{C} \setminus \{0 \}$.
  2. $\Omega = \mathbb{C} \setminus \{P_1,\ldots,P_k \}$.

(Greene and Krantz, Function Theory of One Complex Variable (3rd ed.), Ch 6, Problem 5, rephrased)

My thoughts: For part 1, one may reason as follows. We know that the biholomorphic self maps of $\mathbb{C}$ are the linear transformations $$ f(z) = az+b, \; (a\ne 0).$$ Of these, we only want those that preserve $0$, which means functions of the form $z\mapsto az$.

Edit: Having seen Arthur's remark, there are additional biholomorphic self map in the form $$ z \mapsto \frac{a}{z}, \;\; (a\ne 0).$$

In part 2, trying to generalize to $k$ points, it's clear that the only linear function that preserves even two points is just the identity. But we may find linear functions that permute the set $\{P_1,\ldots,P_k\}$. Any ideas on how to continue ?

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If you consider instead of $\Bbb C$, the Riemann sphere, any biholomorphic function on the sphere, preserving the set $\{0, \infty\}$, would fit (1). You only found half of them. – Arthur Oct 19 '12 at 10:06
@Arthur Thanks. You're right. – Teddy Oct 20 '12 at 17:37
what do you mean by preserve $0$? – Un Chien Andalou May 6 '13 at 4:13
and would you tell me the answer of $2$? – Un Chien Andalou May 6 '13 at 4:14
@Tsotsi: Preserve $0$ means fix $0$, or explicitly $f(0)=0$. The idea to generalize from the first part is that any such biholomorphic map acts as a permutation on $P_1,\ldots,P_k, \infty$. Hence, if $k=2$, one can cook up a Mobius trasformation for each of the 6 permutations. If, however, $k>2$, one cannot generally do so, and the only biholomorphic map would be the identity. – Teddy May 9 '13 at 9:14

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