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Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has n+3 distinct points any n+2 of which are in general position (in the sense of not being contained in an (n-1)-sphere), then f is a Moebius transformation.

Here, we make no any other assumption on f, e.g. continuity, injectivity, surjectivity, and so on. Circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle.

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This theorem is well-known in certain circles. – Ian Agol Dec 30 '12 at 18:28
Can you provide the source of the well-known theorem? – woodbass Dec 30 '12 at 18:35
I imagine this is a follow-up to… – alvarezpaiva Dec 30 '12 at 18:43
Oh, you're not assuming continuous - usually the term "map" refers to a continuous function. I don't know a reference, I think I worked this out when I was a graduate student by showing that it takes spheres to spheres, and induction on dimension. – Ian Agol Dec 30 '12 at 18:45
@Agol: Do you mean that you proved the conjecture under the assumption that f is continuous? If no assumption on continuity, do you have any idea or know any results in literature? – woodbass Dec 30 '12 at 18:58

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