# non linear operator over the mandelbrot set [closed]

I write here because Google Scholar does not give me feedbacks.

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

• 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. – Andreas Blass Feb 11 '19 at 1:03
• Your definition of the Mandelbrot set is incorrect ("finite limit" must be replaced by "bounded"). – Alexandre Eremenko Feb 11 '19 at 13:54
• 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 – 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