0
2013
Impact Factor

# Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

2021, Vol. 17, no. 3, pp.  321-334

Author(s): Barinova M. K., Gogulina E. Y., Pochinka O. V.

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
Citation: Barinova M. K., Gogulina E. Y., Pochinka O. V., Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp.  321-334
DOI:10.20537/nd210306