When calculating Julia Sets and performing iterative operations with real numbers, round-off errors manifest themselves at the boundary of the Julia set. (They may or may not manifest themselves elsewhere, but my efforts were principally at the boundary.)
The boundary is referred to as the Julia set by Kenneth Falconer in Fractals: A Short Introduction (NY: Oxford University Press, 2013). Falconer demonstrates z= z*z+c where c=(0,0) as a nice starting place. It is a nice starting place and the most elementary means of generating a Julia set. Points inside the Julia set converge to (0,0); points outside diverge to infinity; points on the Julia set stay at the unit circle.
However, when performing computer calculations, most the points that should remain on the unit circle do not behave well due to round-off errors. They either run off to infinity (with the postman) or converge to (0,0). The problem is exactly with round-off error.
To convince yourself, open a spreadsheet, and set up a couple of columns for unit circle points, and do the tedious iterative calculations:
Computationally, nothing behaves well on the unit circle.
You can switch to polar coordinates when the starting point c=(0,0). This approach is nice because it shows that points on the unit circle converge piecewise to the unit circle.
Now plot the points for everything I've mentioned above - not the iterations, but the points themselves. It will give you a clear visual perception of the gap between mathematics as it is and what really happens with your computer calculations.