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Math Help - Bilinear transformations

  1. #1
    Member kalyanram's Avatar
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    Bilinear transformations

    Prove that a loxodromic transformation can be expressed as a resultant of elliptic and hyperbolic transformations.

    Thanks and Regards,
    Kalyan.
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  2. #2
    Member kalyanram's Avatar
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    Hi,

    Here is a proof I came up with any bilinear transformation L(\omega, z) with two fixed points say \alpha, \beta will be of the form  \frac{\omega - \alpha}{\omega - \beta} = k.\frac{z - \alpha}{z - \beta} when k is neither unimodular nor  k \in R the transformation is neither elliptical nor hyperbolic in the respective cases, hence loxodromic.

    Now for all  k \in C we know that k = r.(\cos\theta + i\sin\theta) , r \in R and |\cos\theta + i\sin\theta| = 1

    \therefore we can express the transformation L(\omega, z) can be expressed as E(H(\frac{z - \alpha}{z - \beta})) where E(t) = (cis\theta)t and H(t) = r.t , r \in R
    where E(t), H(t) are elliptic and hyperbolic transformations respectively.

    The notation used for E(t), H(t) may be confusing but what I mean is its just a matrix multiplication.

    Let me know if this proof is ok.

    Kalyan.
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