Towards a classification of linear and nonlinear Smale horseshoes

We consider the problem of classification of Smale horseshoes from point of view of the local topological conjugacy of two-dimensionalmaps which generate the horseshoes.We show that there are 10 different types of linear horseshoes. As it was established in the recent paper [4], there are infinitelymany different types of nonlinear horseshoes. All of them belong to the class of the so-called half-orientable horseshoes and can be realized for endomorphisms (not one-to-one maps) of disk or for diffeomorphisms of non-orientable two-dimensional manifolds. We give also a short review of related results from [4].