Wild attractors for Fibonacci maps

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 $d\ge 5.1$).