Total disconnectedness of Julia sets and absence of invariant linefields for real polynomials

In this paper we shall consider real polynomials with one (possibly degenerate) non-escaping critical (folding) point. Necessary and sufficient conditions are given for the total disconnectedness of the Julia set of such polynomials, Also we prove that the Julia sets of such polynomials do not carry invariant linefields. In the real case, this generalises the results by Branner and Hubbard for cubic polynomials and by McMullen on absence of invariant linefields