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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     author = {McKiernan, M.A. and R\'enyi, A.},
     title = {Variational {Aspects} of the {Abel} and {Schroder} {Functional} {Equations}},
     journal = {Canadian mathematical bulletin},
     pages = {257--265},
     year = {1963},
     volume = {6},
     number = {2},
     doi = {10.4153/CMB-1963-023-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1963-023-8/}
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