My question is an amalgamation of two previous questions. The first question I'd like to draw attention to is here. It asks whether there can exist a non trivial semigroup defined on $\mathbb{C}$

$$\phi(s,z):\mathbb{C}_{\Re(s) > 0}\times\mathbb{C} \to \mathbb{C}$$ $$\phi(s_0, \phi(s_1,z)) = \phi(s_0+s_1,z)$$

The answer is no, because such a semigroup would necessarily have a repelling fixed point for some $s_0$, and each function commutes with each other, and an entire function with a repelling fixed point can only have a countable number of functions that commute with it; which forms a contradiction. A big thanks goes to Alexandre Eremenko for referencing I.N. Baker and solving this.

The second question is found here and asks whether an entire function $g$ of finite order can have a composite square-root. The answer is yes. Again, another big thanks goes to Alexandre Eremenko for referencing I.N. Baker again, and solving this.

A great question which hybridizes the above two questions, but laying insoluble in both instances:

For an entire function $f:\mathbb{C} \to \mathbb{C}$, can there exist a sequence of entire functions $\{f_n\}_{n=1}^\infty$ such that: $$f_n(f_n(...(n\,times)...f_n(z))) = f_n^{\circ n}(z) = (f_n \circ f_n \circ ...(n\,times)...\circ f_n)(z) = f(z)$$

This first answer fails to prove this can't happen, this is only a countable list of functions that commute with $f$, so the proof provided by Baker does not work. The second answer suggests this might be possible, but only refers to a square root of $f$, not an arbitrary $n$'th root for all $n$.

Essentially I'm asking this because I've learnt that if $f:\mathbb{C} \to \mathbb{C}$ there is no valid way of constructing

$$(f \circ f \circ...(s\,times)...\circ f)(z) : \mathbb{C}_{\Re(s) > 0}\times\mathbb{C} \to \mathbb{C}$$

so I'm wondering if we can weaken this to, sure there is no semigroup, but there can be

$$(f\circ f\circ... (q\,times)...\circ f)(z):\mathbb{Q} \times \mathbb{C} \to \mathbb{C}$$

My money is on the fact this is impossible and I'm crossing my fingers that Alexandre Eremenko knows why because he's read so much Baker.

  • $\begingroup$ This is a conjecture (perhaps of Baker) which is still unsolved. $\endgroup$ – Alexandre Eremenko Mar 27 '17 at 20:57
  • $\begingroup$ That makes this sound even more intriguing. Is it his conjecture that no such functions $f$ exist other than linear functions, or that there could be such a function? Just to be sure I'm on the winning team. $\endgroup$ – user78249 Mar 27 '17 at 21:11
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    $\begingroup$ The conjecture is that such a function does not exist, except an affine one, so the chance that you loose your money is small. $\endgroup$ – Alexandre Eremenko Mar 27 '17 at 21:20
  • $\begingroup$ @AlexandreEremenko Do you know what some attempts at proving this are? The only way I can think is showing that if $q_k$ is a cauchy sequence of rationals, then $f^{\circ q_k}(z)$ uniformly converges on compact subsets of $\mathbb{C}$, giving a semigroup on an uncountable domain. I think this is rather too simple though, professional attempts are probably more clever than that. $\endgroup$ – user78249 Mar 28 '17 at 0:07
  • $\begingroup$ read Baker's early papers. On compositions of entire functions, his works represent the current state of things, more or less. $\endgroup$ – Alexandre Eremenko Mar 28 '17 at 0:08

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