07/17/2008, 10:48 AM

Code:

`-- What does half-/ fractional-/continuous-/complex iteration mean?`

This problem can only be expressed in terms of the series-paradigm, although

the Reihenalgebra-concept can possibly be seen as an equivalent approach.

The question is - for example -

given a function f(x) = y

what is the function g(x) such that g(g(x)) = y

g(x) is then called the half-iterate of f(x) and is a fractional-iterate

The terms half,fractional and continuous are used if the iterator-parameter

is thought as real, but continuous; if the iterator is thought as a general

complex number, sometimes the term continuous is as well used.

For real iterator h

f°h(x) = f°(n+r)(x) = f°n(f°r(x)) where n is integer and r is fractional

Example using powerseries:

For a function f(x), defined by powerseries, with constant term=0 (f(0)=0) and f'(0)=/=0

it is easy to find the half iterate g(x) by manipulation of the powerseries and

equating coefficients at like powers of x:

Assume f(x) = Ax + Bx^2 + Cx^3 + ...

target g(x) = ax + bx^2 + cx^3 + ...

satisfying g(g(x)) = f(x)

then

g(g(x)) = a g(x) + b g(x)^2 + c g(x)^3 + ...

= a*( ax + bx^2 + cx^3 + ...)

+ b*( ax + bx^2 + cx^3 + ...)^2

+ ...

= a^2 x + (ab + ba^2) x^2 + ...

= f(x) = A x + B x^2 + ...

then by equating coefficents at like powers of x , either a=+sqrt(A) or a=-sqrt(A)

and all other coefficients can then uniquely be determined, so we get

g(x) = sqrt(A) x + B/(sqrt(A) + A)*x^2 + ...

For general fractional iteration-heights the handling of the appropriate powerseries

is much more complicated and suggests the tools of algebra of infinitely-sized matrices.

Formal analytical handling for general functions is much developed and mostly

based on

(see:) Abel - functional relation

Schröder - functional relation

[see : matrix-approach, matrix-logarithm, matrix-diagonalization,

binomial-expansion using functions, ~ using matrix-operators,

function-logarithm (ILog) , exponential polynomial interpolation

<literature>]

[see : Faa di Bruno-formula, ... ]

[see further <literature>: iteration-theory, time-series, dynamical systems]

====================================================================================

Gottfried Helms, Kassel