TITLE: Rounding errors in images of Julia sets
QUESTION [12 upvotes]: One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function.
How do rounding errors affect the results?
I'm looking for references on this issue, especially but not exclusively for the escape time method.
The only ones I have found so far are:

"The [inverse iteration] method is very insensitive to round-off errors, since $f$ tends to be expanding on its Julia set, so that $f^{-1}$ tends to be contracting.'' [Milnor, Dynamics in one complex variable, page 49]
"Theorem 1 makes it plausible that in almost all cases rounding errors do
not affect the computer graphics of Julia sets.'' [Steinmetz, Rational iteration, page 175]

REPLY [5 votes]: 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:
x2-y2, 2.0*x*y
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.
Lynn Wienck