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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

  • $\begingroup$ 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 '19 at 1:03
  • $\begingroup$ Your definition of the Mandelbrot set is incorrect ("finite limit" must be replaced by "bounded"). $\endgroup$ – Alexandre Eremenko Feb 11 '19 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 '19 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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