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Online claude

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(Question) extension of numerical DE bounds to other powers/dimensions?
« on: August 15, 2019, 01:50:50 PM »
The quadratic Mandelbrot distance estimate is \[ d=\frac{1}{\left|\frac{\partial e}{\partial c}\right| \log{2}} \] where e is the normalized continuous escape time.  Then true distance is between d/2 and 2*d, approximately (the interval must be widened slightly).

How does the factor of log(2) change when you change the power of the formula?

How does the factor change when you go into 3D or 4D space?

How do the bounds (d/2, 2*d) change in these cases too?

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Offline hobold

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Re: extension of numerical DE bounds to other powers/dimensions?
« Reply #1 on: August 15, 2019, 02:22:01 PM »
This is a quick and dirty answer without checking back on the literature.

I believe the general factor is log(power), and dimensionality of the surrounding space is not relevant here; where "power" is the highest exponent of the length change of the vector being iterated on.

Edit for clarification:
In triplex coordinates or monopolar spherical coordinates, the amount of "wraparound" (i.e. the rotational exponent) and the amount of length change (i.e. the stretch exponent) can be computed separately. I believe the wraparound is irrelevant to the log(power) factor. But that is really only a guess on my part.

Offline gerrit

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Re: extension of numerical DE bounds to other powers/dimensions?
« Reply #2 on: August 15, 2019, 05:46:36 PM »
Yes log(p) for power p, not sure about bounds. A while ago we found typos in a long paper on this by some Chinese author (did not respond to emails) which proved factor 2, should be possible to look at that paper and see if power can be >2. They also proved formula for quaternion "M-set". Search one of the threads of last year for this.

We had another thread which derived d~1/e' in general (interpret e' in matrix sense) but that's up to a constant.
Proof of log(p) constant and factor 2 accuracy relies on e having analytical properties which is not possible in general I think.

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