" A continuous iteration of f(x,y) r>0 ; f(x,y)^[r] "

[Alain Verghote]

f:R*R->R .Let us start with a simple case:
g(x,g(x,y))= f(x,y) f given (1)
or f(x,y)^[1/2]=g(x,y)
Example: f(x,y)=y/(1+x*y) , g(x,y)=2*y/(2+x*y) verifies (1).

By the same way will have:f(x,y)^[1/p]=p*y/(p+x*y)=g(x,y) p integer
verifying g(x,g(x....)))n times =f(x,y)=y/(1+x*y).
We may generalize:
if f(x,y)=phi^-1(phi(y)+n(x)) or m^[n(x)](y) ,
phi(m(y))=phi(y)+1 ;
then f(x,y)^[r]=phi^-1(phi(y)+r.n(x)) or m^[r.n(x)](y) ,
here r is a positive real number.

Please your comments and ideas,alain.

[Daniel Geisler]

Alain, my tetration.org web site is dedicated to continuous iteration.
See http://www.tetration.org/Dynamics/index.html for resources on
continuous iteration and
http://www.tetration.org/Combinatorics/index.html for an outline of my
combinatorial approach to the subject.

For ideas consider Stephen Wolfram's question:
> How can one extend recursive function definitions to continuous
> numbers? What is the continuous analog of the Ackermann function? The
> symbolic forms of the Ackermann function with a fixed first argument
> seem to have obvious interpretations for arbitrary real or complex
> values of the second argument. But is there a general way to extend
> these kinds of recursive definitions to continuous cases? Given a way
> to do this, how does it apply to recursive definitions like those on
> page 130? What happens to all the irregularities when one is between
> integer values? Or is it only possible to find simple continuous
> generalizations to functions that show fundamentally simple behavior?
> Can this be used as a characterization of when the behavior is simple?

For my non-peer-reviewed response to Wolfram's question on the NKS Forum
see http://forum.wolframscience.com/showthread.php?s=&threadid=488

Here's why I'm interested in the subject. A theory of continuously
iterated smooth matrix functions would probably encompass all of
dynamics in physics. A theory of continuously iterated smooth complex
functions would be powerful enough to extend the Ackermann function to
complex numbers.

Daniel Geisler