...having at least one fixed point.(adsbygoogle = window.adsbygoogle || []).push({});

let g be a given function with fixed point p.

say g is defined on R though it could be C.

here's how we can approximate the fractional iterates of g:

expand the series for the nth iterate of g, denoted g^n, about p.

g^n(p)=p.

(g^n)'(p)=g'(g^(n-1)(p))*g'(g^(n-2)(p))*...*g'(g(p))*g'(p)=g'(p)^n.

IE, the first derivative at p to the nth power.

(g^n)''(p)=(g'(g^(n-1)(x))*g'(g^(n-2)(x))*...*g'(g(x))*g'(x))'|x=p

When you do this and simplify the geometric series, you get

(g^n)''(p)=( g'(p)^(n-1) ((g'(p)^n) - 1) g''(p) ) / ( g'(p)-1).

The third derivative at p is considerably more complex but a CAS can do it easily.

Therefore, a second order approximation to g^n is this:

g^n(x) = g^n(p) + (g^n)'(p)(x-p) + (1/2)(g^n)''(p)(x-p)^2+O(x-p)^3.

g^n(x) = p + g'(p)^n(x-p) +

(1/2)( ( g'(p)^(n-1) ((g'(p)^n) - 1) g''(p) ) / ( g'(p)-1) ) ) (x-p)^2 + O(x-p)^3.

These formulas enable us to let n be a fraction or any number.

Example:

g(x)=e^x. The desire is to find a semi-iterate or so called "half exponential" function h such that h(h(x))=g(x); in the above, this would be accomplished by taking n to be 1/2.

Let p be a (complex) fixed point of g. FIx a branch of the square root function; ie make p^(1/2) consistant throughout.

g^n(x) = p + g'(p)^n(x-p) +

(1/2)( ( g'(p)^(n-1) ((g'(p)^n) - 1) g''(p) ) / ( g'(p)-1) ) ) (x-p)^2 + O(x-p)^3, ok?

Well, g'(x)=e^x and g''(x)=e^x, so g'(p)=g''(p)=p. So we have:

g^n(x) = p + p^n(x-p) +

(1/2)( ( p^(n-1) ((p^n) - 1) p) / ( p-1) ) ) (x-p)^2 + O(x-p)^3.

If n=1/2...

h(x)=p + SQRT(p)(x-p) + (1/2)( ( (SQRT(p) - 1) SQRT(p)) / ( p-1) ) ) (x-p)^2 + O(x-p)^3.

THere you have it, a half iterate of e^x to second order. I have a feeling that for real x this will all turn out real but I'm not sure.

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# Fractional iterates of analytic functions

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