On classification of classical and half-orientable horseshoes in terms of boundary points

Recently, Smale horseshoes of new types, the so called half-orientable horseshoes, were found in [1]. Such horseshoes may exist for endomorphisms of the disk and for diffeomorphisms of non-orientable two-dimensional manifolds as well.They have many interesting properties different from those of the classical orientable and non-orientable horseshoes. In particular, half-orientable horseshoes may have boundary points of arbitrary periods. It is shown from this fact that there are infinitely many types of such horseshoes with respect to the local topological congugacy. To prove this and similar results, an effective geometric construction is used.