A Case of Nonexistence of Generalized Pseudo-Anosov Homeomorphisms With One Needle On a Closed Nonorientable Surface of Genus 3

In this paper, we prove the nonexistence of a generalized pseudo-Anosov homeomorphism on a closed nonorientable surface of genus 3 whose invariant foliations have two singularities, one of which has valency 1 and the other has valency 5. The proof essentially uses the construction of the so-called band surface of a generalized pseudo-Anosov homeomorphism for which a number of conditions imposed on the structure of its boundary are formulated. The considered set of singularities is not the only admitted by the Euler – Poincaré formula for genus 3 and the number of valency 1 singularities being equal to 1. Thus, the statement which we prove provides a case demonstrating difference from orientable surfaces on which there is a generalized pseudo-Anosov homeomorphism with any admitted set of singularities if at least one of them is of valency 1, except for the case of torus for which any sets of singularities with a single singularity of valency 1 are impossible.