Abstract: On the basis of considering the likely limit sets of a 4-order Feigenbaum's map and their Hausdorff dimension, we have proved that for any t∈(0,log_{3^1/2+12), there always exits such a 4-order nonsingle-valley Feigenbaum’s map which has a likely limit set with Hausdorff d imension t.}

Key words: Feigenbaum’s map, likely limit sets, Hausdorff dimension, simple periodic orbits