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The Mandelbrot set M could be defined as the set of all the complex plane point c where the recurrent sequences $z_{n+1} = z_nz_n+c$ and $z_0=c$ have finite limit.

I was wondering if we apply a general operator $L$ that transforms the sequence $z_n$ into a new one if that this affects, in general, the Mandelbrot set.

For example if $L(z_n) = z_n + 1$ then we get that if $z_n$ diverge so will do $L(z_n)$. Then $L$ applied to M will only obtain a new manderbrot set only shifted in the complex plain. Are there some general results on these subject?

Thanks Paolo


closed as unclear what you're asking by YCor, Andreas Blass, Ben McKay, Alexandre Eremenko, Lasse Rempe-Gillen Feb 11 at 19:35

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  • $\begingroup$ Applying the transformation $L(z_n)=z_n+1$ to every $z_n$ is very difference from applying $L$ to $M$, becaus e$L$ doesn't preserve the relationship $Z_{n+1}={z_n}^2+c$ between consecutive $z$'s. $\endgroup$ – Andreas Blass Feb 11 at 1:03
  • $\begingroup$ Your definition of the Mandelbrot set is incorrect ("finite limit" must be replaced by "bounded"). $\endgroup$ – Alexandre Eremenko Feb 11 at 13:54
  • $\begingroup$ right but this does not change the original question that is if a recurrent sequence is bounded again we can ask if applying a non-linear operator to that sequence preserve it's boundness so that if $z_n$ acting in a complex point $c$ that is bounded and so is in Mandelbrot set remains bounded in its image and so the point $c$ is still in mandelbrot set $\endgroup$ – pgiacome Feb 11 at 14:52

Sorry for the notation you're right but what I mean is

if $c=0$ the recurrent sequence $z_{n+1} = z_{n}^2 + c$ is $z_{n} = \{0,0,0,\cdots\}$ so it converge then $c=0 \in M$. If we apply the operator $L(z_{n+1})=z_{n+1} + 1$ then we have that the recurrent sequences generate from $c=0$ becomes the following one $L(z_{n+1})= \{1,1,\cdots\}$ that converge.

So basically every point $c \in M$ have the same behavior on its recurrent sequence begin that if

$\lim_{n \rightarrow \infty} z_n$ converge than the same will do $\lim_{n \rightarrow \infty} L(z_n)$.

This is a simple case but if we move to more complex operator (i.e. non-linear ones) that relation could not hold. Meaning that if a point $c$ is $\in M$ then the same point could not be in $M$.

So my question is there are some line of reasearch on this subject?

Thans Paolo


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