Geodesic
Omega-classification of Surface Diffeomorphisms
Russian journal of nonlinear dynamics, Tome 17 (2021) no. 3, pp. 321-334

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The present paper gives a partial answer to Smale's question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}^{} \subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.
Keywords: Smale diagram, (A,B)-diffeomorphism, $\Omega$-conjugacy. 
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     author = {M. K. Barinova and E. Y. Gogulina and O. V. Pochinka},
     title = {Omega-classification of {Surface} {Diffeomorphisms}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {321--334},
     publisher = {mathdoc},
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     number = {3},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2021_17_3_a5/}
}
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M. K. Barinova; E. Y. Gogulina; O. V. Pochinka. Omega-classification of Surface Diffeomorphisms. Russian journal of nonlinear dynamics, Tome 17 (2021) no. 3, pp. 321-334. http://geodesic.mathdoc.fr/item/ND_2021_17_3_a5/