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

V. Grines; D. Mints

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arXiv · 2111.11895

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

V. Grines·D. Mints

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Abstract

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 surfaces such that their non-wandering set 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 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.

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V. Grines, D. Mints. 2021-11-23. On decomposition of ambient surfaces admitting A-diffeomorphisms with nontrivial attractors and repellers. https://arxiv.org/abs/2111.11895

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