Geodesic
Variational Aspects of the Abel and Schroder Functional Equations
Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 257-265

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Given an analytic function f, the successive iterates of f are defined byf[0](z) = z, f[n+1](z) = f{f[n](z)} for every z.In particular f[1](z) = {f[0](z)} f(z). Extensive study has been given [1] to the problem of generalizing the iterates f[n], for integer n, to f[t] for arbitrary t, where the iterative character of f[t] is to be preserved by the conditions,f[0](z) = z and f[s]{f[t](z)} = f[s+t](z) for arbitrary s and t. 
McKiernan, M.A.; Rényi, A. Variational Aspects of the Abel and Schroder Functional Equations. Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 257-265. doi: 10.4153/CMB-1963-023-8
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[1] 1. See for example, Szekeres, G., Regular Iteration of Real and Complex Functions, Acta Mathematica, 100 (1958), pp. 203-258. Also P. Erdős and E. Jabotinsky, On Analytic Iteration, Journal D'Analyse Mathematique, Vol. 8, part 2, 1960/61, Jerusalem.10.1007/BF02559539 Google Scholar

[2] 2. Schrőder, E., Uber iterierte Funktionen, Math. Ann., 2(1870) pp. 317-365. Google Scholar

[3] 3. The equations are given in tensor form for example in Sokolnikoff, I. S., Tensor Analysis, John Wiley and Sons, 1951, p. 161. Google Scholar

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