  Existence of wild attractors -- attractors whose basin has a positive
Lebesgue measure but is not a residual set -- has been one of central themes in
one-dimensional dynamics. It has been demonstrated by H. Bruin et al. that
Fibonacci maps with a sufficiently flat critical point admit a wild attractor.
We propose a constructive trichotomy that describes possible scenarios for the
Lebesgue measure of the Fibonacci attractor based on a computable criterion. We
use this criterion, together with a computer-assisted proof of existence of a
Fibonacci renormalization $2$-cycle for non-integer critical degrees, to
demonstrate that Fibonacci maps do not have a wild attractor when the degree of
the critical point is $d=3.8$ (and, conjecturally, for $2< d \le 3.8$), and do
admit it when $d=5.1$ (and, conjecturally, for $d \ge 5.1$).
