Anatoly Medvedev

In this paper, we prove the theorem given in the title and a similar theorem for generalized pseudo-Anosov homeomorphisms of the Klein bottle with only one “needle” singularity. We reformulate both statements in terms of the invariant foliation singularity type and orientability of foliations. The method we use to prove the theorems implies using the construction of band surface. Every band surface is represented with its combinatorial description, called the configuration. Applying a series of Rauzy transformations to all possible configurations in the cases considered, we show that the necessary conditions imposed on the invariant foliations are violated whence the results follow.

In this paper, we provide a complete answer to the question posed in its title. For all closed surfaces for which this question has not yet been resolved, namely, closed nonorientable surfaces of odd genus greater than 5, a series of examples of pseudo-Anosov homeomorphisms are constructed. Each of these homeomorphisms is defined by means of the so-called code, which implies using the construction of a band surface. Combined with previously obtained results, this allows us to fill in the gap and to formulate a general result: pseudo-Anosov homeomorphisms exist on a closed nonorientable surface if and only if its genus is at least 4.