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Wolfram as well as Aldrovandi and Freitas 1 maintain that iterated functions $f^t(x)$ are a valid alternative to PDEs for modelling physics. Instead of just citing 1, I want to be able to cite the initial paper and author who justified using iterated functions in physics. I do not mean simply modeling a problem in physics, I mean modeling physics itself. I study the mathematics and structure of $f^t(x)$. It is my contention that if $f^t(x)$ has universal properties, then those properties must occur in physics.

Schroeder wrote Über iterirte Functionen, the first paper on dynamical systems in 1871, but this appears to be a paper of pure mathematics. Poincare is the first person I think used the dynamical systems of iterated functions to study physics.

R. Aldrovandi and L. P. Freitas, Continuous iteration of dynamical maps, J. Math. Phys. 39, 5324 (1998)

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    $\begingroup$ As written, this question appears to be too broad to be answered in the concise Q&A format that is the SE model. $\endgroup$ – Kyle Kanos Apr 1 '14 at 12:29
  • $\begingroup$ @KyleKanos I have narrowed the question down. $\endgroup$ – Daniel Geisler Apr 1 '14 at 12:42
  • $\begingroup$ Hmm, my guess is that it'd go back to Newton or so (Newton-Raphson root-finding is an iterative method). Is there a particular reason you want the first and don't want to cite the above? $\endgroup$ – Kyle Kanos Apr 1 '14 at 13:06
  • $\begingroup$ What do you mean by "modeling physics itself". We model problems or problem domains, but surely not physics as a whole ? $\endgroup$ – StephenG Apr 24 '17 at 15:05
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    $\begingroup$ Strictly physics... not population biology (salmon), for instance? I'd doubt if one could work backwards past the RG. $\endgroup$ – Cosmas Zachos Apr 24 '17 at 17:02
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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.

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  • $\begingroup$ I suspect that the physics of iterated functions has been around for longer that the 1950's, but I think your paper is an important move forward, it shows actual advantages of using iterated functions in physics. This is the first work in my knowledge that does so. $\endgroup$ – Daniel Geisler May 4 '17 at 4:19
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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.

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  • $\begingroup$ Perhaps a little too straightforwardly? For a constant Hamiltonian this is the explicit Lie group, so, then merely functionally composing $f(x)=x^s$ so $f^t (x)=x^{st}$? $\endgroup$ – Cosmas Zachos Apr 24 '17 at 12:26
  • $\begingroup$ Well, the way I had in mind was $x\cong |\psi\rangle$ and $f^t(x)\cong U(t)|\psi\rangle$, so that $f^t(f^s(x))\cong U(t)U(s)|\psi\rangle = U(t+s)|\psi\rangle \cong f^{t+s}(x)$. In your terminology, setting $x= U(\Delta t)$ and setting $f(x)=x^s=U(s\Delta t)$, implies that $f^t$ just increments time forward in units of $s\Delta t$. $\endgroup$ – TotallyRhombus Apr 24 '17 at 15:03
  • $\begingroup$ Correct; this is just a Lie Group action of a line, not exactly a self-similar dynamical system. $\endgroup$ – Cosmas Zachos Apr 24 '17 at 15:09
  • $\begingroup$ Well, then specialize 'quantum mechanics' to 'conformal field theory'. $\endgroup$ – TotallyRhombus Apr 24 '17 at 16:52

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