[FOM] Fractional Iteration and Rates of Growth

Dmytro Taranovsky dmytro at MIT.EDU
Tue May 9 19:26:06 EDT 2006

```Joe Shipman wrote:
> why would there be non-uniqueness [for fractional iteration]
> in the case where your base function
> is 2^x instead of 2*x?

Uniqueness of fractional iterates is based on analytic f having a
fix-point 'a' of a positive multiplier, f(a)=a and f'(a)>0, combined
with the restriction to analytic fractional iterates g with g(a)=a and
g'(a)>0.  It does not hold in general; for example,  x/(x-1) is a
half-iterate of y = x.

> If there is a real analytic function defined on the positive real axis,
> with f(1)=1 and f(x+1)=2^f(x), and if, as you say, such a function
> could not have all of its derivatives be non-negative at any point,
> then it is not a satisfactory extension of the "tower" function, since
> we surely want such an extension to have the property of "all
> derivatives positive everywhere".

The foundational issue here is what counts as a natural extension of,
for example, the exponential tower function.  Such extension should be
analytic, defined for all positive real numbers, and satisfy the
functional equation.  However, "all derivatives positive everywhere" is
too strong and should be weakened to: for every n, as x approaches
infinity, the nth derivative approaches infinity.

Ideally, we would have a uniqueness result matching the foundational
requirements.

Dmytro Taranovsky
```