06/19/2009, 08:51 AM

I just discovered that this was already researched in 1968 [1]. Karlin and Mcgregor showed that:

If is a function holomorphic and single valued on the complement of a closed countable set in the extended complex plane. Let two fixed points of such that and . Then the regular iterations at and are equal if and only if is a fractional linear function.

[1] Karlin, S., & Mcgregor, J. (1968). Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Am. Math. Soc., 132, 137–145.

If is a function holomorphic and single valued on the complement of a closed countable set in the extended complex plane. Let two fixed points of such that and . Then the regular iterations at and are equal if and only if is a fractional linear function.

[1] Karlin, S., & Mcgregor, J. (1968). Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Am. Math. Soc., 132, 137–145.