# Tetration Forum

Full Version: The mystery of 2 fixpoints.
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Hello tetration freaks !

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.

However if $f([s_1,s_2])\subseteq [s_1,s_2]$ is not true , can we have regular iterations that are independant of the fixpoint used ?

regards

tommy1729