06/19/2009, 08:51 AM
I just discovered that this was already researched in 1968 [1]. Karlin and Mcgregor showed that:
If \( f \) is a function holomorphic and single valued on the complement of a closed countable set in the extended complex plane. Let \( s_1\neq s_2 \) two fixed points of \( f \) such that \( |f'(s_0)|,|f'(s_1)|\neq 0,1 \) and \( f([s_1,s_2])\subseteq [s_1,s_2] \). Then the regular iterations at \( s_1 \) and \( s_2 \) are equal if and only if \( f \) 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 \( f \) is a function holomorphic and single valued on the complement of a closed countable set in the extended complex plane. Let \( s_1\neq s_2 \) two fixed points of \( f \) such that \( |f'(s_0)|,|f'(s_1)|\neq 0,1 \) and \( f([s_1,s_2])\subseteq [s_1,s_2] \). Then the regular iterations at \( s_1 \) and \( s_2 \) are equal if and only if \( f \) 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.