# Difference between revisions of "Regular iterate"

m (Text replacement - "\$([^\$]+)\$" to "\\(\1\\)") |
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− | [[Regular iterate]] of some function |
+ | [[Regular iterate]] of some function \(T\), referred below as a [[transfer function]], at its [[fixed point]] \(L\) is such [[iterate]] \(T^n\) that is regular at \(L\) even at non–integer values of \(n\). |

− | In particular, for integen numbers |
+ | In particular, for integen numbers \(m\) and \(n\!\ne\!0\), the [[regular iterate]] \(f=T^{m/n}\) is supposed to be [[fractional iterate]] of function \(T\), id est, for \(z\) in vicinity of point \(L\), |

− | (1) |
+ | (1) \( ~ ~ ~ f^n(z)=T^m(z)\) |

− | The regular iterate can be evaluated with [[regular iteration]] of the asymptotic expansion of the [[Abel function]] |
+ | The regular iterate can be evaluated with [[regular iteration]] of the asymptotic expansion of the [[Abel function]] \(G\) in vicinity of \(L\) and corresponding expansion of the [[superfunciton]] \(F=G^{-1}\), |

and iterative application of the [[Transfer equation]] in order to bring the argument of the [[superfunciton]] to the range of values where the asymptotic expansion provides the required precision. |
and iterative application of the [[Transfer equation]] in order to bring the argument of the [[superfunciton]] to the range of values where the asymptotic expansion provides the required precision. |
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## Revision as of 18:25, 30 July 2019

Regular iterate of some function \(T\), referred below as a transfer function, at its fixed point \(L\) is such iterate \(T^n\) that is regular at \(L\) even at non–integer values of \(n\).

In particular, for integen numbers \(m\) and \(n\!\ne\!0\), the regular iterate \(f=T^{m/n}\) is supposed to be fractional iterate of function \(T\), id est, for \(z\) in vicinity of point \(L\),

(1) \( ~ ~ ~ f^n(z)=T^m(z)\)

The regular iterate can be evaluated with regular iteration of the asymptotic expansion of the Abel function \(G\) in vicinity of \(L\) and corresponding expansion of the superfunciton \(F=G^{-1}\), and iterative application of the Transfer equation in order to bring the argument of the superfunciton to the range of values where the asymptotic expansion provides the required precision.

## Keywords

Iteration of function, Superfunciton, Abel function, Abel equation, Schroeder equation, Schroeder function, Zooming equation, Zooming function