bo198214 Wrote:But if you had this lemma, how would you continue with the uniqueness of tetration?

First, I repeat my Lemma: Let

is entire 1-periodic funcrion

for some

Then

(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

and let

within the range of holomorphism of tetration, id est, in the complex plane except some set of measure zero.

This set includes one line

, and, at

,

additional horizontal cut lines that correspond to the periodicity of tetration.

Assume there exist some function

which is also holomorphic within some region

,

which includes at least some vicinity of the segment

.

Assume also, that function

is holomorphic at least in

and

and

is not tetration

. (Tetration

satisfies the equations above)

Then, there exist function

such that in vicinity of the segment

function

can be expressed as follows:

and, in some vicinity of the same segment,

We need this expression only in vicinity of the segment

; therefore, we have no need to specify, which of

we mean.

There exist only one holomorphic extension of a function in a domain of trivial topology.

Therefore, we can extend the function

outside of the range of definition.

Consider function

Assuming that

and

are not identical in the segment

, function

should not be identically zero. This function also allows the holomorphic extension. Consider behavior of function

in the complex plane. This extension should be periodic, satisfying conditions of my Lemma.

Therefore, function

should take values from set

inside domain

. Function

has singularities at these points.

Therefore, function

has singularities in the domain

. With these singulatiries,

function

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

.