A remark on generalized iterates

Abstract

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx eZ, letf |n| denote then'th iterate off. Clearlyf |m|(f |n|(x))=f |m+n|(x) for allm,neZ and allxeR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxeR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F |α|:αeG} on the real line with the properties (i)F |α|(F |β|(x))=F |α+β| (x) for allxeR and (ii)F |n|(x)=f |n|(x) for allneZ andxeR. 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.