On decomposition of ambient surfaces admitting A-diffeomorphisms with non-trivial attractors and repellers

Discrete and Continuous Dynamical Systems. 2022. Vol. 42. No. 7. P. 3557–3568.

Grines V., Mints D.

It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $A$ and the topology of the ambient manifold. In the given article, this statement is considered for the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable connected surfaces, the non-wandering set of each of which consists of $k_f\geq 2$ connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism $f\in \mathbb G(M^2)$ is homeomorphic to the connected sum of $k_f$ closed orientable connected surfaces and $l_f$ two-dimensional tori such that the genus of each surface is determined by the dynamical properties of appropriating connected component of a basic set and $l_f$ is determined by the number and position of bunches, belonging to all connected components of basic sets. We also prove that every diffeomorphism from the class $\mathbb G(M^2)$ is $\Omega$-stable but is not structurally stable.

Keywords: базисное множество, A-diffeomorphism, А-диффеоморфизм, неблуждающее множество, basic set, one-dimensional attractor and repeller, non-wandering set, одномерные аттрактор и репеллер

Publication based on the results of: Бифуркации в семействах диффеоморфизмов и связь топологии многообразия с динамикой, заданной на нем системы (2021)