[FOM] 605: Integer and Real Functions (Harvey Friedman)

[Timothy Y. Chow]

On Fri, 28 Aug 2015, Joe Shipman wrote:
> Is there f with f(f(x))=e^x which is both real analytic and monotonic 
> for x>0?

Yes, Kneser's function is injective.

> Is Kneser's construction generalizable to find monotonic real analytic 
> g(g(x))=f(x) and so on?
>
> Is it generalizable to find monotonic real analytic f(f(x))=2^x?
>
> What kind of conditions on such f imply uniqueness?

I don't know the answers to these questions off the top of my head.  On 
the page that I linked to before http://reglos.de/lars/ffx.html there is a 
reference to a book "Iterative Functional Equations" by Kuczma et al. that 
I know addresses these kinds of questions.  Partial answers are known but 
there tend to be a lot of hypotheses and conditions on the theorems, 
making the subject rather difficult to digest.  I'm not sure if this is 
intrinsic to the subject or a reflection of our lack of ability to prove 
things.

Tim