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Article

Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 3. P. 321-334.
Barinova M., Гогулина Е. Ю., Pochinka O.

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 Ω-conjugated are constructed. Moreover, a subset G∗ ⊂ G of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient Ω-conjugacy is singled out.