Mandelbrot set is defined using the complex equation $P_c (z)=z^2 +c$ where $c$ is a complex number
Let Set $M=${$(c,k,m) |$ the sequence $P_c (0),P_c (P_c (0)), P_c (P_c (P_c (0)))...$ is unbounded and some number $P^n_c (0)$ has a complex magnitude greater than 2 (but less than $m$) for some integer $n$ > $k$}
Assume $k$ and $m$ are integers given in unary encoding
Is there an efficient algorithm in the real computing model (i.e the Blum-Shub-Smale model) for deciding set $M$ ?
Essentially, I'm asking if there exists efficient algorithm that decides whether the sequence is UNBOUNDED such that breakaway occurs at a number of iterations more than K.
UPDATE: The problem is in $NP$ since there is a polynomially verifiable certificate of membership.
So, Is there an efficient algorithm or is it $NP$-hard to decide set $M$?
The problem went through major changes for these reasons:
1-For the problem to be interesting, "<" was changed to ">" in the definition.
2-I wanted the problem to be in $NP$, so I introduced parameter $m$.
Also, the bounty may be awarded to the first person who poses an $NP$-complete problem related to Mandelbrot Set