backward stability in the Mandelbrot set

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Offline claude

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« on: August 26, 2018, 03:23:54 PM »
Following up from these posts:
https://fractalforums.org/fractal-mathematics-and-new-theories/28/perturbation-theory/487/msg2365#msg2365
https://fractalforums.org/fractal-mathematics-and-new-theories/28/are-the-mandelbrot-sets-generated-different-in-appearance-to-the-actual-set/1333/msg6800#msg6800

This paper references Fatou 1919:
https://www.academia.edu/16246867/BACKWARD_STABILITY_FOR_POLYNOMIAL_MAPS_WITH_LOCALLY_CONNECTED_JULIA_SETS
Quote
In the present paper we study the problem of backward stability which is an extension of the Lyapunov (forward) stability to the backward dynamics of a non-invertible map. Essentially, this notion was first introduced by Fatou who showed that f : C → C is backward stable at points not belonging to the limit sets of critical points. Other facts concerning backward stability which follow from classical results (in particular, from the description of the local dynamics at periodic points given, e.g., in [CG]) are that f  : C→ C is not backward stable at any parabolic periodic point which lies in the Julia set. Obviously, f is not backward stable at attracting periodic points. Thus, the well-known obstacle for the backward stability of a polynomial at a point is that the point could be an attracting or neutral periodic point.

The relevant Fatou paper I think is http://archive.numdam.org/article/BSMF_1919__47__161_0.pdf but it's 100+ pages in French, so I haven't read it.

However, this may all be irrelevant, as this is for f:C → C, which works for Julia sets, but in the Mandelbrot set we're curious about backward stability wrt 'c' due to rounding errors in 'z', so the interesting f : CxC → C,


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