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+--- 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