Geodesic
Solenoidal representations and the~homology of hyperbolic attractors of diffeomorphisms of surfaces
Sbornik. Mathematics, Tome 188 (1997) no. 6, pp. 799-821

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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. 
@article{SM_1997_188_6_a0,
     author = {A. Yu. Zhirov},
     title = {Solenoidal representations and the~homology of hyperbolic attractors of diffeomorphisms of surfaces},
     journal = {Sbornik. Mathematics},
     pages = {799--821},
     publisher = {mathdoc},
     volume = {188},
     number = {6},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_6_a0/}
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A. Yu. Zhirov. Solenoidal representations and the~homology of hyperbolic attractors of diffeomorphisms of surfaces. Sbornik. Mathematics, Tome 188 (1997) no. 6, pp. 799-821. http://geodesic.mathdoc.fr/item/SM_1997_188_6_a0/