The mystery of 2 fixpoints. - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: The mystery of 2 fixpoints. (/showthread.php?tid=856) The mystery of 2 fixpoints. - tommy1729 - 05/03/2014 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