The solution of Feigenbaum’s functional equation is called Feigenbaum’s map. We investigate the likely limit set (i.e., the maximal attractor in the sense of Milnor) of non-unimodal Feigenbaum’s map, estimate the Hausdorff dimension of the likely limit set for the non-unimodal Feigenbaum’s map, and for every  construct a non-unimodal Feigenbaum’s map which has a likely limit set with Hausdorff dimension s.

non-unimodal Feigenbaum’s map, attractor, likely limit set, Hausdorff dimension.