Examples of one-dimensional hyperbolic attractors on nonorientable surfaces (Q1569790)

This paper deals with some problems related to the existence of diffeomorphisms generating hyperbolic attractors on nonorientable surfaces. From the point of view of the ``realizability'' of an attractor on a surface, here attractors are studied up to topological conjugacy of their generating diffeomorphisms on the attractor neighborhoods. An order relation on the set of closed compact surfaces is introduced, for defining the minimal surface of the attractor of a diffeomorphism. The attractor is the union of the unstable manifolds of its points. The definitions of bunch, degree of a bunch, orientability of an attractor on surfaces are recalled. Noting that the attractor realization depends on its boundary type, and the orientability, the basic question arises as to what boundary types of orientable and nonorientable attractors, formally permissible by the Euler-Poincaré formula for a given surface, are actually realizable. The paper is devoted to the nonorientable case, which remainded open. Examples of diffeomorphisms with attractors permit to prove four theorems.