Can you give me an example of a homeomorphism of the extended complex plane that is not circle-preserving?
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Homeomorphism of what? – markvs Aug 23 '20 at 02:31
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Wow I'm so sorry. I have forgotten to write that. – nomeaning Aug 23 '20 at 02:37
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What do you mean by "cycle preserving"? Is a hyperbolic geodesic line a cycle? Is it "complex plane" or the "upper half plane"? – markvs Aug 23 '20 at 02:41
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It is circle-preserving. It maps circles into circles. The circles of the extended complex plane are in two form: Either it is a Euclidean circle in the complex plane or it is a Euclidean line in the complex plane combined with the point at infinity. – nomeaning Aug 23 '20 at 02:44
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The extended complex plane is the union of the complex plane joined with the point at infinity. – nomeaning Aug 23 '20 at 02:45
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So you are not talking about hyperbolic plane. – markvs Aug 23 '20 at 02:49
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I know only one model of the hyperbolic plane that is the upper-half plane of the complex plane. – nomeaning Aug 23 '20 at 02:55
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There are several other models (the two disc models, for example). – markvs Aug 23 '20 at 02:57
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The linear map given by the matrix $$\left(\begin{array}{cc}2 & 1\\ 1 & 1\end{array}\right)$$ is a homeomorphism which does not preserve circles.
markvs
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Its determinant is nonzero so that makes it a Möbius transformation and Möbius transformations preserve circles. Do I miss something? – nomeaning Aug 23 '20 at 02:53
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1Moebius transformation corresponding to this matrix would be $\frac {2z+1}{z+1}$. My answer is about the linear map $a+bi\to 2a+b+(a+b)i$ – markvs Aug 23 '20 at 02:56
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Thank you for your answer. I'm sorry that I can not upvote your answer because I'm new here. – nomeaning Aug 23 '20 at 03:02
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