Geodesic
Topological classification of M\"obius transformations
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 175-183

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Linear fractional transformations on the extended complex plane are classified up to topological conjugacy. Recall that two transformations $f$ and $g$ are called topologically conjugate if there exists a homeomorphism $h$ such that $g=h^{-1}\circ f\circ h$, in which $\circ$ is the composition of mappings. 
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     author = {T. V. Rybalkina and V. V. Sergeichuk},
     title = {Topological classification of {M\"obius} transformations},
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T. V. Rybalkina; V. V. Sergeichuk. Topological classification of M\"obius transformations. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 175-183. http://geodesic.mathdoc.fr/item/FPM_2012_17_6_a4/