Fractal Related Discussion > Fractal Mathematics And New Theories

(Question) Inverse iteration

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gerrit:
I understand inverse iteration for quadratic Julia set of z^2 + c: inverse iterate from any point z0 in Julia set and you get a tree of ancestors which generates the whole Julia set J.

What about the filled Julia set K without boundary, with c inside the M-set but not on boundary.
Does inverse iteration from any point in K also reach all points in K? Can't find this in the literature.

marcm200:
When I start an inverse iteration from the point 0.10+0.13*i inside the largest Fatou component of the basilica z^2-1 and mark the inversed points with yellow, I get the image below which looks remarkably like inverse iteration from the Julia set boundary (with some spattered points in said Fatou component).

So maybe the inverse iteration from a point in the interior produces a set that clusters around the boundary?  I was surprised to see that it does not fill out any area part of the interior itself. All due to rounding errors using sqrt? But it would make sense, as when you use inverse iteration from a boundary point, due to limited precision, points inversely followed are usually not exactly on the boundary but close - inside or outside, so from that point of view it's not that surprising that the two starting sets produce similar outputs I guess.

(This was just a quick implementation which was not checked thouroughly, but it also produces solely a circle when using the tamest, non-fractal c=0).

pauldelbrot:
This is to be expected. The Julia set is a repeller ordinarily; inverse iteration turns it into an attractor. The general result is a tree of preimages accumulating to the Julia set itself, so very few interior points "far" from the boundary will show up; or very few exterior points, if starting out there.

gerrit:

--- Quote from: marcm200 on June 18, 2021, 10:13:58 AM ---When I start an inverse iteration from the point 0.10+0.13*i inside the largest Fatou component of the basilica z^2-1 and mark the inversed points with yellow, I get the image below which looks remarkably like inverse iteration from the Julia set boundary (with some spattered points in said Fatou component).

So maybe the inverse iteration from a point in the interior produces a set that clusters around the boundary?  I was surprised to see that it does not fill out any area part of the interior itself.

--- End quote ---

--- Quote from: pauldelbrot on June 18, 2021, 04:06:00 PM ---This ["does not fill out any area part of the interior itself" I assume, G] is to be expected. The Julia set is a repeller ordinarily; inverse iteration turns it into an attractor. The general result is a tree of preimages accumulating to the Julia set itself, so very few interior points "far" from the boundary will show up; or very few exterior points, if starting out there.

--- End quote ---
I was thinking that any point in K converges to the periodic attractor, so inverse iteration from any z0 on the stable attractor should generate the whole K (interior of filled Julia set).

Reason I'm asking is I try to figure out what the equivalence classes as defined by 't Hooft in his (free) book https://link.springer.com/content/pdf/10.1007%2F978-3-319-41285-6.pdf of a quadratic Julia set are.

z and w are in the same equivalence class if there exists an N>0 (# iters) such that z and w iterate to the same point after precisely N iters. So what are the equivalence classes of z^2 + c (c inside M)? I thought the exterior, Julia set and interior K and no more. But if inverse iteration does not generate K when z0 is in K that's not it.