The problem of existence of a strange attractor for degree 2 critical circle covers whose dynamics on the post-critical set is semi-conjugated to the golden mean rotation (so called Fibonacci covers) has been extensively studied. The method is based on the drift of a Markov map of Keller et al, which remains the only approach known to try to decide this kind of question.

It was shown that as the degree of criticality tends to infinity, the drift tends to a finite limit and the sign of this limit decides whether there is a strange attractor or an acim. While at present there is no plan to estimate the sign analytically, the method is sufficiently constructive to allow at least a credible numerical estimate. It turns out that the drift is negative and hence no strange attractor.

The setting of the problem and results known so far will be reminded and then I will present the constructive numerical approach.

(based largely on joint work with G. Levin and G. Siudem)