Question: 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

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> Is there an efficient algorithm in the real computing model (i.e the Blum-Shub-Smale model) for deciding set $M$ ?
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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.

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> So, Is there an efficient algorithm or is it $NP$-hard to decide set $M$?
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**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**

Answer: 
This answer combines some of my comments to the question and improves them.

The set M is decidable in linear time on a Turing machine over the reals.

Here is why. The definition of the set M in the question consists of two conditions, but note that the first condition is redundant: it is well-known that the infinite sequence Pc(0), Pc(Pc(0)), Pc(Pc(Pc(0))), … is unbounded if it contains a term whose absolute value is greater than 2. Therefore, to decide whether a given pair (c,k) belongs to the set M or not, we only have to test the second condition, which can be done in O(k) time on a Turing machine over the reals. Since k is given in unary, this is linear time in the input length.

I do not know much about Turing machines over the reals, and I may have made any mistakes.
This problem related to Mandelbrot set seems to be $NP$-complete:

$M=${$(c,k,r) |$ In the sequence $P\_c (0),P\_c (P\_c (0)), P\_c (P\_c (P\_c (0)))...$ of first $k$ complex numbers, there is a subset $T$ of complex numbers such that the sum of the real parts $\gt$ $r.k$ and the sum of imaginary parts $\gt$ $r.k$}

$r$ is real number and $k$ is an integer in unary.

Here is a geometric interpretation, since each $P\_c^i(0)$ is a vector in 2D, we want to find the maximum size square obtainable by the summation of a subset of two dimensional vectors.

A promising reduction is from (optimization) number partition problem where each partition has the same number of elements (in the optimization version, we minimize the difference between the two sums).