# Is there some general technique to turn the operation of function into a continuous transformation from input to output?

To be precise:

Given a function $$f(x)$$ I want a function $$g(t, x)$$, $$x,t \in \mathbb{R}$$ or $$\mathbb{C}$$, such that $$g(t, x)$$ equals the $$t$$-time application of $$f$$ to the value $$x$$ for $$t \in \mathbb{N}$$,

This means, $$g(0, x) = x, g(1, x) = f(x), g(2, x) = f(f(x))$$, and g should be continuous in $$t$$.

It feels like I want something like an "analytic continuation", like the gamma function extending the discrete factorial function into a smooth transformation. Is there some algorithm that can be applied to get some kind of taylor expansion or even a closed form systematically?

For example,

• if $$f(x) = x+c$$, then I want $$g(t, x) = x+ct$$

• if $$f(x) = ax$$, then $$g(t, x) = a^t x$$

• if $$f(x) = ax+b$$, then $$g(t, x) = a^t x + \frac{b(a^t-1)}{a-1}$$

the last one I managed to figure out the closed form for, but e.g. for $$f(x) = x^2 + c$$ the iterated function explodes into increasingly higher degree polynomials that I don't know how to handle.

My formal understanding of calculus is on the level of standard university courses, so if this goes way way deeper, please be gentle!

• $g$ is the iterated function: en.wikipedia.org/wiki/Iterated_function
– IV_
Commented May 26, 2023 at 14:47
• Unfortunately, this does go way deeper than calculus. fix $x$, which is irrelevant at this point. The recursion defines $g$ for integer values of $t$ only. Now choose any continuous function $h$ defined on $[0,1]$ with $h(0) = x, h(1) = f(x)$. and define $g(t,x) = h(t)$ for $t \in [0,1]$. The recursion then extends $g$ to a continuous function for all $t$. But a different function for each $h$ you pick. This over-abundance of choices for $g$ remains even if you require $g$ to be infinitely differentiable. To get a "best" choice, you need to require that $g$ be analytic. Commented May 27, 2023 at 15:59
• (cont) That means $g$ can be represented by a power series about each point $t$. Complex analysis has tools for building analytic functions from a sequence of points such as $g(n, x)$, but I'm rusty on the results, and will leave a fuller explanation to others. Commented May 27, 2023 at 16:04
• Thanks for the comments! I should have stumbled onto the wiki page myself, apparently what I want is a "fractional iterate" or "flow". Okay so I guess its a much deeper rabbit hole than I thought. I was just hoping that this could be used to "interpolate" smooth trajectories for e.g. tracking the orbit of a point acted on by a dynamical system. Then I'll probably just use something spline-like, even though it's not showing the "correct" path... Commented May 28, 2023 at 15:34