Explicit examples of functions with flow?

Question

Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept flow (I think?). I don't understand the Wikipedia-article on this subject very well, though. I was hoping for some nice, concrete examples of iterated functions extended to the real or even complex numbers with which I might understand things better. If we take $f(x) = x^2 +3$, for example, what would $f^{\sqrt(2)}(x)$ be? Or, even more ambitiously, say that $g(x)=e^x$ How do we find $g^{\pi^2 + 3i}(x)$?

Thanks,

Max Muller

Editorial to the moderators: perhaps this should be CW?

Answer 1

I thought the wikipedia article is pretty straightforward. Define a function $\phi: \mathbb{R}^2 \to \mathbb{R}: (x,t) \mapsto \phi(x,t)$. Now, you want the second parameter $t$ to be interpreted as the number of times you have applied the function to $x$, in a way. To formalize this, you introduce the following rule on $\phi$:

$$\phi(\phi(x,t),s)=\phi(x,t+s)$$

and this for all $x,t$ and $s$. In particular, you see that:

$$\phi(\phi(x,t),t)=\phi(x,2t)$$

and more generally if we define $\phi_t:\mathbb{R}\to\mathbb{R}:x\mapsto\phi(x,t)$

$$\phi_t^n(x)=\phi(x,nt)=\phi_{nt}(x)$$

which is exactly the behaviour you would like to have.

Determining a flow $\phi(x,t)$ starting from the condition that $\phi(x,1)=f(x)$ is not an easy task however and can not be done for any arbitrary function I think. I remember another post related to the question. I think it was a question about the Vieta product.

Answer 2

Max, besides the existence of a lot of literature about this subject I've written two very basic/introductive articles about it. Perhaps the better is the shorter (and newer) one, I deal with the iteration of $ f(x) = 1/(1+x) $ The idea stems originally from a consideration of continued fractions in a general form $ x; 1/(1+x); 1/(1+1/(1+x));....$ . On the other hand, it is simple and the iteration of the function can be approached from different viewpoints, so the approaches are crosschecked mutually. See : fractional functional iteration of f(x)=1/(1+x)

additional comment: the difficulties of defining a continuous iteration of $f(x) = e^x$ compared with that of a polynomial $f(x)=x^2 + 3$ are so eminent that putting them into the same question requires a much too wide area for discussion in a MO-thread[Edit: Oh sorry, we're not in MO, I was jumping between tabs in my browser, sorry]. However in the last years a lot of material, including scans of historical articles, was put online and can be read. Possibly I can provide useful links if you ask more specific questions. Also you might browse the tetration-forum at Tetration-forum There is also an online reference-database. see tet-forum

[Update] i constructed a function f(x,h) where f(x,0)=x, f(x,1)=x^2+3 and f(x,h) the h'th iterate using fixpoint-shift and matrix-diagonalization. Unfortunately this gives complex powerseries for fractional iterates. The construction gives the following complex plot for x=1,h=0..2 . note that for real, noninteger heights h the curve extends into the complex plane! So I think, there should be a better solution...

Answer 3

Since you ask for explicit examples, the solution to any time-invariant ordinary differential equation as a function of time and the initial value of the variable is a flow. For example, $dx/dt = \lambda x$ with initial value $x(0) = x_0$ has the solution $f(t,x_0) = e^{\lambda t} x_0$, which in your notation can be interpreted as $f^t(x) = e^{\lambda t} x$. This is a somewhat boring example; the interesting ones happen in more than one dimension. Rotation, for example: let $f^t(x)$ be the rotation of $x$ about a fixed axis by an angle $\omega t$. I'll leave it to you to think of more complicated ODEs.