# What is the event in history where iterated functions became appropriate for modeling physics?

You are effectivelly talking about the Renormalization group in quantum field theory, the paradigmatic self-similar system underlying the whole of nature--with ultimate (subsequent) dramatic consequences in the strong interactions.

Specifically, in 1954, Gell-Mann and Low introduced their eponymous RG equation for QED, $$g(\mu)=G^{-1}\left(\left(\frac{\mu}{M}\right)^d G(g(M))\right),$$ for some function G (Wegener's function, the QFT version of Schroeder's function, see Appendix B of their paper. T D Lee is thanked.) and a constant d, in terms of the coupling g(M) at a reference scale M.

Gell-Mann and Low, ostensibly unaware of Schroeder's equation, nevertheless exploited the extraordinary functional composition properties of this result: they realized that the effective scale can be arbitrarily taken as μ, and can thus vary to define the theory at any other scale, as well, $$g(\kappa)=G^{-1}\left(\left(\frac{\kappa}{\mu}\right)^d G(g(\mu))\right) = G^{-1}\left(\left(\frac{\kappa}{M}\right)^d G(g(M))\right).$$

This, in effect amounts to arbitrary iteration of the map connecting the couplings at two disparate scales, with the iteration index t, in your language and the contemporary QFT parlance, being the continuous logarithm of the scale M. In effect, gs are functionally conjugated through G to multiplications of scale ratios.

This paradigm actually illustrates your point: physicists would rather integrate the infinitesimal differential version of this equation in perturbation theory (β function), rather than solve the recondite Schroeder equation here, ab initio. Nevertheless, my collaborator and I have, in fact, done just that, to indicate that it is possible: Curtright, T. L.; Zachos, C. K. (March 2011). "Renormalization Group Functional Equations". Physical Review D. 83 (6): 065019.

How about just plain old quantum mechanics? There, the time evolution map (or operator) satisfies $f^t (f^s(x))=f^{t+s}(x)$ straightforwardly. The important 'universal properties' of the time evolution operator are unitarity and various physical symmetries depending on context. In classical physics, unitarity is essentially replaced by Liouville's theorem. Unitarity is at the heart of many important results in quantum mechanics and also quantum field theory, and its precise role in quantum gravity is what the information paradox is about.