RUS  ENG
Full version
JOURNALS // Russian Journal of Nonlinear Dynamics // Archive

Rus. J. Nonlin. Dyn., 2021 Volume 17, Number 3, Pages 321–334 (Mi nd759)

Mathematical problems of nonlinearity

Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

M. K. Barinova, E. Y. Gogulina, O. V. Pochinka

National Research University Higher School of Economics, ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia

Abstract: 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.

MSC: 37D05

Received: 14.07.2021
Accepted: 07.09.2021

Language: english

DOI: 10.20537/nd210306



Bibliographic databases:


© Steklov Math. Inst. of RAS, 2024