TITLE: Is there a effective computational criterion to all periodic points of a rational function are repelling.
QUESTION [6 upvotes]: I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a rational function, which is related to Fatou's  theorem.
I tried some examples , however, sometimes I can not find attracting periodic point (period from 2 to 100) with the comupters.  I have no idea to know whether it has a attracting basin basin for $f$ or not. For example, $$f(z) = -2/3*z+(-2*z^3+1)/(3*z^3+5*z).$$ By the software of Ultra Fractal, and it seems that the julia set of the example is very small (may be a Cantor set), I can only see the whole screen is black.

REPLY [3 votes]: The best way to study these questions is in the framework of computable analysis. I will take your rational function as being given by oracles for its coefficients and for its critical points. (That is, for a given epsilon, we expect to be able to know these values up to an error of epsilon.)
In this framework, the question "Does f have non-repelling periodic points?" is undecidable. Indeed, suppose that the rational map is such that it has only repelling periodic points and is not structurally stable. (Conjecturally, every map having only repelling periodic points is structurally unstable. Under many circumstances this is known; the simplest case is the postcritically finite case where it follows from work of Thurston.) 
Then there are rational maps arbitrarily nearby that do have nonrepelling cycles, and hence we cannot distinguish between the two cases by considering our map only up to finite precision.
However, if we replace "non-repelling" by "attracting", then the problem is semi-decidable. Indeed, we basically just iterate the critical values and see whether they converge to an attracting cycle; this can be done rigorously. (I will omit the details, but suspect they can be found e.g. in the work of Braverman and Yampolsky.)
Of course, when your coefficients are given in some absolute way (e.g. as rational or algebraic numbers), then you can ask about this as a classical problem in computer science. This problem seems less natural, although I would expect it also to be undecidable. There is a result that says that the Mandelbrot set is undecidable in the sense of Shub-Smale (which assumes infinite-precision arithmetic), which suggests at least that the problem can't be solved in any naive way.
Of course, there are specific examples for which it can be verified that there are no nonrepelling cycles, as mentioned by Alexandre.