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

Vyacheslav Grines; Dmitrii Mints

doi:10.3934/dcds.2022024

Article Contents

2022, Volume 42, Issue 7: 3557-3568. Doi: 10.3934/dcds.2022024

This issue Previous Article On a class of singularly perturbed elliptic systems with asymptotic phase segregation Next Article Particle approximation of one-dimensional Mean-Field-Games with local interactions

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

Vyacheslav Grines and
Dmitrii Mints^,

HSE University, Bolshaya Pecherskaya 25/12, Nizhny Novgorod, Russia, 603155

^* Corresponding author: Dmitrii Mints
^* Corresponding author: Dmitrii Mints

Received: August 2021

Revised: January 2022

Early access: March 2022

Published: July 2022

This work was financially supported by the Russian Science Foundation (project 21-11-00010), except for the proofs of Lemma 3.2 and Theorem 2. The proof of Lemma 3.2 was obtained with the financial support from the Academic Fund Program at the HSE University in 2021-2022 (grant 21-04-004). The proof of Theorem 2 was obtained with the financial support from the Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant (ag. 075-15-2019-1931)

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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 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.

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Mathematics Subject Classification: 37D20.

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Figure 1. Phase portrait of the diffeomorphism a) $ f_1 $; b) $ f_2 $

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Figure 2. Construction of the characteristic curve for the bunch of degree 4

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Figure 3. Construction of the surface $ M_{\Lambda} $

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