On Which Closed Nonorientable Surfaces Do Pseudo-Anosov Homeomorphisms Exist?

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.