Uniqueness of tetration. Small lemma.

[Kouznetsov]

But if you had this lemma, how would you continue with the uniqueness of tetration?

First, I repeat my Lemma: Let

\( S = \{ z \in \mathbb{C} {\text \bf ~:~} |\Re(z)| {\!\le\!} 1 \} \)

\( M = \{ n \in \mathbb{N} {\text \bf ~:~} n {\!\le\!} -2 \} \)

\( h \) is entire 1-periodic funcrion

\( h\big(z^{*}\big) = h(z)^{*} ~ \forall~ z ~\in \mathbb{C} \)

\( h(0) = 0 \)

\( h(x) \ne 0 \) for some \( x \in \mathbb{R} \)

\( J(z) = z+h(z) ~~~ \forall~ z ~\in \mathbb{C} \)

Then \( \exists t \in S {\text \bf ~:~} J(t)\in M \)

(end of Lemma 1).

This lemma does not requre any knowledge about properties of tetration.

Now, how do I apply it:

Assume some fixed base \( b\!>\!0 \) and let \( F(z)=\mathrm{tet}_{b}(z) \) within the range of holomorphism of tetration, id est, in the complex plane except some set of measure zero.
This set includes one line \( z {\!\le\!}-2 \), and, at \( 1 \!<\! b \!<\! \exp(1/\mathrm{e}) \),
additional horizontal cut lines that correspond to the periodicity of tetration.

Assume there exist some function \( G(z) \) which is also holomorphic within some region \( s \),
which includes at least some vicinity of the segment
\( |\Re(z)| \! {\!\le\!} \! 1 \).

Assume also, that function \( f=G \) is holomorphic at least in \( \{z\in \mathbb{C} {\text \bf ~:~ \Re(z)>-2 \} \) and

\( f(z+1)=\exp\big( f(z) \big) \)

\( f(0)=1 \)

\( f\big(z^{*}\big)=f(z)^{*} \)

\( f^{\prime}(x) > 0 \forall x> -2 \)

and \( G \) is not tetration \( F \). (Tetration \( F \) satisfies the equations above)

Then, there exist function \( J \) such that in vicinity of the segment
\( [-1,1] \) function \( G \) can be expressed as follows:
\( G=F\big(J(z)\big) \)
and, in some vicinity of the same segment,
\( J(z)=F^{-1}\big( G(z) \big) \)

We need this expression only in vicinity of the segment \( [-1,1] \); therefore, we have no need to specify, which of \( G^{-1} \) we mean.

There exist only one holomorphic extension of a function in a domain of trivial topology.
Therefore, we can extend the function \( J \) outside of the range of definition.
Consider function
\( h(z)=J(z)-z \)
Assuming that \( F \) and \( G \) are not identical in the segment \( [-1,1] \), function \( h \) should not be identically zero. This function also allows the holomorphic extension. Consider behavior of function \( h \) in the complex plane. This extension should be periodic, satisfying conditions of my Lemma.
Therefore, function \( J \) should take values from set \( M \) inside domain \( S \). Function \( F \) has singularities at these points.
Therefore, function \( G \) has singularities in the domain \( S \). With these singulatiries,
function \( G \) does not satisfy the criterion in the definition of tetration.
In such a way, there exist only one tetration, that is strictly increasing function in the segment \( [-1,1] \).