**MSC2010** 05C15

### The existence connected characteristic space at the gradient-like diffeomorphisms of surfaces

#### E. V. Nozdrinova^{1}

Annotation | In this paper we consider the class $G$ of orientation preserving gradient-like diffeomorphisms $f$ defined on a smooth oriented closed surfaces $M^2$. Establishes that for any such pair of diffeomorphisms there is a dual attractor-repeller $A_f,R_f$, which have a topological dimension not greater than 1 and the orbit space in their Supplement $V_f$ is homeomorphic to the two-dimensional torus. The immediate consequence of this result is the same period of separatrices of a saddle of all diffeomorphisms $f\in G$. A lot of many classification results for structurally stable dynamical systems with a non-wandering set consisting of a finite number of orbits --- Morse-Smale systems --- is based on the possibility of such representation for the dynamics of a system in the form ``source-sink''. For example, for systems in dimension three, there always exists a connected characteristic space associated with the choice of a one-dimensional dual attractor-repeller pair. In dimension two this is not true even in the gradient-like case, however, in this paper it will be shown that there exists a one-dimensional dual pair whose characteristic orbit space is connected. |
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Keywords | gradient-like diffeomorphisms, attractor, repeller |

^{1}**Elena V. Nozdrinova**, laboratory assistant, Laboratory of topological methods in dynamics, National Research University Higher School of Economics (25/12 Bolshaja Pecherskaja Str., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/ 0000-0001-5209-377X, maati@mail.ru

**Citation**: E. V. Nozdrinova, "[The existence connected characteristic space at the gradient-like diffeomorphisms of surfaces]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:2 (2017) 91–97 (In Russian)

**DOI** 10.15507/2079-6900.19.201701.091-097