By János Aczél (auth.), János Aczél (eds.)
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8) Functional analysis. Iteration of a function can be discussed by (crudely speaking) considering the linear operator of right composition with the function. This leads to new insights and results. (9) Phantom iterates. Since functions in general have no iterative roots (fractional iterates) of every order (and thus no continuous time iterate) "generalized embeddings" have been sought for a long time. Since 1984, the idea of phantom iterates offers one such approach. ( 10) An Appendix briefly discusses various topics which are outside the main fields outlined above but should be mentioned in our survey.
Reich and J. Schwaiger using the L (2) (3) (4) (5) (6) normal forms under conjugation in r in particular the smooth normal forms for 1. The proofs also furnished the construction of the iteration groups and pointed to connections with autonomous differential systems and differential equations with complex linearization, following P. Erd6s and E. Jabotinsky. Analogous results (using the normal forms mentioned above) were found for the existence of iterative roots. From these criteria grew results on connection between existence of analytic iterations and of "sufficiently many" iterative roots of an FEr (results of L.
No. 69 Prace Mat. 9 (1979), 73 -83. LIEDL, R. , Die Losung der Translationsgleichung mittels schneller Pilgerschritttransformation. Grazer Math. Berichte 314 (199 I), I-54. , Sur les decompositions invariantes du demi-groupe du groupe abelien lineairement ordonne. Wyz. Szkola Ped. Krak6w. -Dydakt. No. 69 Prace Mat. 11 (1985), 119-145. , Some class of solutions of the translation equation (Polish). Kieleckie Studia Mat. 5 (1989), 23-55. , On an extension of solution of the translation equation from a subsemigroup onto the group.