Geodesic
On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers
Sbornik. Mathematics, Tome 188 (1997) no. 4, pp. 537-569

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Necessary and sufficient conditions for topological conjugacy are established in the case of structurally stable, orientation-preserving diffeomorphisms of a two-dimensional smooth closed oriented manifold $M$ that belong to the class $S(M)$, that is, satisfy the following conditions: 1) all the non-trivial basic sets of each $f\in S(M)$ are one-dimensional attractors or repellers; 2) there exist only finitely many heteroclinic trajectories lying in the intersections of stable and unstable manifolds of saddle periodic points belonging to trivial basic sets. 
@article{SM_1997_188_4_a1,
     author = {V. Z. Grines},
     title = {On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers},
     journal = {Sbornik. Mathematics},
     pages = {537--569},
     publisher = {mathdoc},
     volume = {188},
     number = {4},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_4_a1/}
}
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V. Z. Grines. On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers. Sbornik. Mathematics, Tome 188 (1997) no. 4, pp. 537-569. http://geodesic.mathdoc.fr/item/SM_1997_188_4_a1/