Difference between revisions of "Regular iterate"

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[[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 $n$.
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[[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 \(n\).
   
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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 $L$,
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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 \(L\),
   
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(1) $ ~ ~ ~ f^n(z)=T^m(z)$
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(1) \( ~ ~ ~ f^n(z)=T^m(z)\)
   
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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]] $F=G^{-1}$,
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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 [[superfunction]] \(F=G^{-1}\),
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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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and iterative application of the [[Transfer equation]] in order to bring the argument of the [[superfunction]] to the range of values where the asymptotic expansion provides the required precision.
   
 
==Keywords==
  
==Keywords==
⚫
[[Iteration of function]],
  
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[[Superfunciton]],
  
⚫
[[Abel function]],
  
 
[[Abel equation]],
  
[[Abel equation]],
  ⚫
[[Abel function]],
  ⚫
[[Iteration of function]],
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[[Superfunction]],
 
[[Schroeder equation]],
  
[[Schroeder equation]],
 
[[Schroeder function]],
  
[[Schroeder function]],
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[[Zooming function]]
  
[[Zooming function]]
   
  ⚫
[[Category:Abel function]]
  ⚫
[[Category:English]]
 
[[Category:Iteration]]
  
[[Category:Iteration]]
 
[[Category:Superfunction]]
  
[[Category:Superfunction]]
⚫
[[Category:Abel function]]
  
 
[[Category:Transfer function]]
  
[[Category:Transfer function]]
 
[[Category:Transfer equation]]
  
[[Category:Transfer equation]]
⚫
[[Category:Zooming equation]]
  
 
[[Category:Schroeder equation]]
  
[[Category:Schroeder equation]]
 
[[Category:Schroeder function]]
  
[[Category:Schroeder function]]
 
[[Category:Zooming function]]
  
[[Category:Zooming function]]
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[[Category:Articles in English]]
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[[Category:Zooming equation]]

Latest revision as of 06:42, 20 July 2020

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 superfunction \(F=G^{-1}\), and iterative application of the Transfer equation in order to bring the argument of the superfunction to the range of values where the asymptotic expansion provides the required precision.

Keywords

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

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