For some other values of c, z_{0} = 0 does not belong
to a cycle, but some of its iterates belong to a cycle. For example, consider c = 2. 
z_{0} = 0, 
z_{1} = z_{0}^{2} + c = 0^{2}  2 = 2, 
z_{2} = z_{1}^{2} + c = (2)^{2}  2 = 2, 
z_{3} = z_{2}^{2} + c = 2^{2}  2 = 2, 
and so z_{n} = 2 for all n > 1. 

That is, z_{0} = 0 is not itself fixed, but it iterates to
a point (z_{2} = 2 in this case) that is fixed. 
Points c with this property are called Misiurewicz points. 
All Misiurewicz points belong to the boundary of the Mandelbrot set. 
They are branch tips, centers of spirals, and points where branches meet. 
Some Misiurewicz points are indicated by red dots. 

The Misiurewicz points are scattered throughout the boundary of
M: every circle centered at every boundary point encloses infinitely many Misiurewicz
points. 