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==Fractional iterates and flows, and negative iterates==
[[File:TrivFctRootExm svg.svg|thumb|{{color|#20b080|''g'': '''R'''→'''R'''}} is a trivial functional 5th root of {{color|#901070|2=''f'': '''R'''<sup>+</sup>→'''R'''<sup>+</sup>, ''f''(''x'') = sin(''x'')}}. The computation of ''f''({{frac|π|6}}) = {{frac|1|2}} = ''g''<sup>5</sup>({{frac|π|6}}) is shown.]]
The notion {{math|''f''{{i sup|1/''n''}}}} must be used with care when the equation {{math|1=''g''<sup>''n''</sup>(''x'') = ''f''(''x'')}} has multiple solutions, which is normally the case, as in [[Functional square root|Babbage's equation]] of the functional roots of the identity map. For example, for {{math|1=''n'' = 2}} and {{math|1=''f''(''x'') = 4''x'' − 6}}, both {{math|1=''g''(''x'') = 6 − 2''x''}} and {{math|1=''g''(''x'') = 2''x'' − 2}} are solutions; so the expression {{math|''f''<sup> 1/2</sup>(''x'')}} does not denote a unique function, just as numbers have multiple algebraic roots
Fractional iteration of a function can be defined: for instance, a [[functional square root|half iterate]] of a function {{mvar|f}} is a function {{mvar|g}} such that {{math|1=''g''(''g''(''x'')) = ''f''(''x'')}}.<ref>{{cite web |work=MathOverflow |title=Finding f such that f(f(x))=g(x) given g |url=https://mathoverflow.net/q/66538 }}</ref> This function {{math|''g''(''x'')}} can be written using the index notation as {{math|''f''<sup> 1/2</sup>(''x'')}} . Similarly, {{math|''f''<sup> 1/3</sup>(''x'')}} is the function defined such that {{math|1=''f''<sup>1/3</sup>(''f''<sup>1/3</sup>(''f''<sup>1/3</sup>(''x''))) = ''f''(''x'')}}, while {{math|''f''{{i sup|2/3}}(''x'')}} may be defined as equal to {{math|''f''{{i sup| 1/3}}(''f''{{i sup|1/3}}(''x''))}}, and so forth, all based on the principle, mentioned earlier, that {{math|1=''f''<sup> ''m''</sup> ○ ''f''<sup> ''n''</sup> = ''f''<sup> ''m'' + ''n''</sup>}}. This idea can be generalized so that the iteration count {{mvar|n}} becomes a '''continuous parameter''', a sort of continuous "time" of a continuous [[Orbit (dynamics)|orbit]].<ref>{{cite journal |first1=R. |last1=Aldrovandi |first2=L. P. |last2=Freitas |title=Continuous Iteration of Dynamical Maps |journal=J. Math. Phys. |volume=39 |issue=10 |pages=5324 |year=1998 |doi=10.1063/1.532574 |arxiv=physics/9712026 |bibcode=1998JMP....39.5324A |hdl=11449/65519 |s2cid=119675869 |hdl-access=free }}</ref><ref>{{cite journal |first1=G. |last1=Berkolaiko |first2=S. |last2=Rabinovich |first3=S. |last3=Havlin |title=Analysis of Carleman Representation of Analytical Recursions |journal=J. Math. Anal. Appl. |volume=224 |pages=81–90 |year=1998 |doi=10.1006/jmaa.1998.5986 |doi-access=free }}</ref>
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