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The Mandelbrot Set is generated by iterating $f(z)=z^2+c$.

Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions?

Likewise, is there a "half-half-step" (or "quarter-step") function $h(z)$ such that

$$h(h(z))=g(z)$$

and

$$h(h(h(h(z))))=f(z)$$

?

If so can we subdivide the steps to infinity? i.e.

$$g_n(g_n(z))=g_{n-1}(z)$$

and

$$g_0(z)=f(z)$$

?

And could we thereby define a continuation of the Mandelbrot iteration function?

spraff
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  • https://math.stackexchange.com/questions/208996/half-iterate-of-x2c – Claude Oct 16 '21 at 14:32
  • @Claude the answer to that question in particular is related but doesn't exactly answer this. It doesn't describe what a half iterate should be at all, or whether one even exists (which it may not) – Max Nov 24 '21 at 22:39
  • @Claude I don't think the linked question reasonably answers this one. The maths might be too complex for OP, and they don't directly talk about the problem of fractional iterates of x^2+c – Max Nov 24 '21 at 23:21
  • Do you want to make continuous dynamical systems from discrete dynamical systems? – Adam Mar 19 '22 at 21:31
  • @Adam Yes that's part of what I'm wondering about. – spraff Mar 20 '22 at 10:27
  • https://www.worldscientific.com/doi/abs/10.1142/S0218127411028295 – Adam Mar 20 '22 at 17:41
  • https://math.stackexchange.com/questions/2579773/do-two-exponential-spirals-intersect – Adam Mar 20 '22 at 17:42

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