  We show that polarized endomorphisms of rationally connected threefolds with
at worst terminal singularities are equivariantly built up from those on Q-Fano
threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are
obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a
consequence, we show that every smooth Fano threefold with a polarized
endomorphism of degree > 1, is rational.
