Orientability of invariant foliations of pseudo-Anosovian homeomorphisms and branched coverings

This paper deals with pseudo-Anosov homeomorphism (possibly generalized) of closed orientable surface; invariant foliations of this mapping are supposed to be non-orientable. A construction is described that helps to build two-sheeted (in general, branched) covering of this surface and pseudo-Anosov homeomorphism of the covering surface that covers the original and has orientable invariant foliations. The covering and the homeomorphism are constructed by means of original mapping. If the original homeomorphism does not have singularities of odd valency then the constructed covering is not branched. Otherwise it has branch points of multiplicity 2 in singularities of odd valency. It is established that in the first case the covering homeomorphism has twice the number of singularities of the same valences as compared with the original. In the second case the number of singularities of even valencies doubles and the features of doubled odd valences are added to them.