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