Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

We consider a family of strongly-asymmetric unimodal maps [Formula: see text] of the form [Formula: see text] where [Formula: see text] is unimodal, [Formula: see text] , [Formula: see text] is of the form and [Formula: see text] where we assume that [Formula: see text] . We show that such a family contains a Feigenbaum–Coullet–Tresser [Formula: see text] map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the [Formula: see text] map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.