Solenoidal representations and the homology of hyperbolic attractors of diffeomorphisms of surfaces

For an arbitrary connected one-dimensional hyperbolic attractor of a diffeomorphism of a closed surface (orientable or not), representations are constructed in the form of generalized solenoids generated by maps of one-dimensional complexes. The construction leads to the determination of such a representation from the union of any finite number of periodic orbits contained in the attractor. Furthermore, the number $m$ of zero-dimensional simplexes of the complex obtained is equal to the number of periodic points chosen, and the number of one-dimensional simplexes is determined by this $m$ and by the so-called boundary type of the attractor. As an application, the one-dimensional Alexandroff–Cech integral homology group of the attractor is computed. The rank of this group is also determined by the boundary type of the attractor.