A remark on generalized iterates |
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Authors: | R C Mullin |
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Institution: | (1) Department of Combinatorics and Optimization, University of Waterloo, N2L 3G1 Waterloo, Ont., Canada |
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Abstract: | Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf
|n| denote then'th iterate off. Clearlyf
|m|(f
|n|(x))=f
|m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F
||:G} on the real line with the properties (i)F
||(F
||(x))=F
|+|
(x) for allxR and (ii)F
|n|(x)=f
|n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday |
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Keywords: | Primary 39B05 |
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