Editing Iterated function (section)

==Lie's data transport equation==
{{see also|Shift operator#Functions of a real variable}}
Iterated functions crop up in the series expansion of combined functions, such as  {{math|''g''(''f''(''x''))}}.

Given the [[Koenigs function#Structure of univalent semigroups|iteration velocity]], or [[beta function (physics)]],  
:<math>v(x) = \left. \frac{\partial f^n(x)}{\partial n} \right|_{n=0}</math> 
for the {{mvar|n}}<sup>th</sup> iterate of the function {{mvar|f}}, we have<ref>{{Cite journal | last1 = Berkson | first1 = E. | last2 = Porta | first2 = H. | doi = 10.1307/mmj/1029002009 | title = Semigroups of analytic functions and composition operators | journal = The Michigan Mathematical Journal | volume = 25 | pages = 101–115 | year = 1978 | doi-access = free }}  {{Cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1088/1751-8113/43/44/445101 | title = Chaotic maps, Hamiltonian flows and holographic methods | journal = Journal of Physics A: Mathematical and Theoretical | volume = 43 | issue = 44 | pages = 445101 | year = 2010 | arxiv = 1002.0104 | bibcode = 2010JPhA...43R5101C | s2cid = 115176169 }}</ref>
:<math>
g(f(x)) = \exp\left[ v(x) \frac{\partial}{\partial x} \right] g(x).
</math>
For example, for rigid advection, if {{math|''f''(''x'') {{=}} ''x'' + ''t''}}, then {{math|''v''(''x'') {{=}} ''t''}}.  Consequently,  {{math|''g''(''x'' + ''t'') {{=}} exp(''t'' ∂/∂''x'')  ''g''(''x'')}}, action by a plain [[shift operator]].

Conversely, one may   specify {{math|''f''(''x'')}}  given an arbitrary {{math|''v''(''x'')}}, through the generic [[Abel equation]]  discussed above,
:<math>
f(x) = h^{-1}(h(x)+1) ,
</math>
where
:<math>
h(x) = \int \frac{1}{v(x)} \, dx .
</math>
This is evident by noting that 
:<math>f^n(x)=h^{-1}(h(x)+n)~.</math>

For continuous iteration index {{mvar|t}}, then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,
:<math>e^{t~\frac{\partial ~~}{\partial h(x)}} g(x)= g(h^{-1}(h(x )+t))= g(f_t(x)).</math>
The initial flow velocity {{mvar|v}} suffices to determine the entire flow, given this exponential realization which automatically provides the  general solution to the ''translation functional equation'',<ref  name="acz">Aczel, J. (2006), ''Lectures on Functional Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, {{ISBN|978-0486445236}}.</ref>  
:<math>f_t(f_\tau (x))=f_{t+\tau} (x)  ~.</math>