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Let's consider a function $ f: \widehat{\Bbb C} \to \widehat{\Bbb C}$ ( riemman sphere) such that:

$i)$ $ T\left( {\frac{1} {{\overline z }}} \right) = \frac{{ - 1}} {{\overline {T\left( z \right)} }} $

$ii)$ T has two fixed points.

My question is if it's true that T is an automorphism of $\widehat{\Bbb C}$. I suspect that it's a rotation (for the antipodals) But I'm not sure. And I want to know if it's true, in that case proving this (it's an automorphism ) only by algebraic manipulations.

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    Do you really want a $-1$ in the numerator on the right, and a $+1$ in the numerator on the left?2012-10-22

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