If a map $f:\mathbb{C}_\infty \to \mathbb{C}_\infty$ preserves all cross ratios, is it necessarily a Mobius map?
Is any map preserving cross ratios a Mobius map?
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complex-analysis
mobius-transformation
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1By "preserves all cross ratios", you mean for example $$\operatorname{CR}\bigl(f(z), f(0), f(1), f(\infty)\bigr) = \operatorname{CR}(z,0,1,\infty)$$ for all $z$? – 2017-01-11
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0I mean $\operatorname{CR}(f(a),f(b),f(c),f(d)) = \operatorname{CR}(a,b,c,d)$ for all $a,b,c,d \in \mathbb{C}_\infty$. – 2017-01-11
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0Yes, now specialise $b,c,d$ and think about what you know about cross ratios. – 2017-01-11