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July  2022, 42(7): 3557-3568. doi: 10.3934/dcds.2022024

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

Received  August 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

Fund Project: 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)

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: A-diffeomorphism, basic set, one-dimensional attractor and repeller, non-wandering set.
Mathematics Subject Classification: 37D20.
Citation: Vyacheslav Grines, Dmitrii Mints. On decomposition of ambient surfaces admitting $ A $-diffeomorphisms with non-trivial attractors and repellers. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3557-3568. doi: 10.3934/dcds.2022024
References:
[1]

S. Kh. Aranson and V. Z. Grines, The topological classification of cascades on closed two-dimensional manifolds, Russian Math. Surveys, 45 (1990), 1-35.  doi: 10.1070/RM1990v045n01ABEH002322.

[2]

S. Kh. Aranson and V. Z. Grines, Cascades on surfaces, in Dynamical Systems IX (eds. D. V. Anosov), Springer, (1995), 141–175. doi: 10.1007/978-3-662-03172-8_3.

[3]

S. Kh. Aranson, R. V. Plykin, A. Yu. Zhirov and E. V. Zhuzhoma, Exact upper bounds for the number of one-dimensional basic sets of surface A-diffeomorphisms, Journal of Dynamical and Control Systems, 3 (1997), 1-18.  doi: 10.1007/BF02471759.

[4]

R. Bowen, Periodic points and measures for Axiom a diffeomorphisms, Transactions of the American Mathematical Society, 154 (1971), 377-397.  doi: 10.2307/1995452.

[5]

V. Z. Grines, The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. I(in Russian), Tr. Mosk. Mat. Obs., 32 (1975), 35-60. 

[6]

V. Z. Grines, On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers, Sb. Math., 188 (1997), 537-569.  doi: 10.1070/SM1997v188n04ABEH000216.

[7]

V. Z. Grines, On topological classification of A-diffeomorphisms of surfaces, Journal of Dynamical and Control Systems, 6 (2000), 97-126.  doi: 10.1023/A:1009573706584.

[8]

V. Z. Grines and Kh. Kh. Kalai, Diffeomorphisms of two-dimensional manifolds with spatially situated basic sets, Russian Uspekhi Math. Surveys, 40 (1985), 221-222. 

[9]

V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2-and 3-Manifolds, Springer, 2016. doi: 10.1007/978-3-319-44847-3.

[10]

V. Z. Grines, O. V. Pochinka and S. van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749.  doi: 10.17323/1609-4514-2016-16-4-727-749.

[11]

M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.

[12] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge university press, 1997. 
[13]

J. Palis, On the $C^1$ $\Omega$-stability conjecture, Publ. Math. IHES, 66 (1988), 211-215. 

[14]

R. V. Plykin, The topology of basis sets for Smale diffeomorphisms, Math. USSR-Sb., 13 (1971), 297-307.  doi: 10.1070/SM1971v013n02ABEH001026.

[15]

R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Math. USSR-Sb., 23 (1974), 233-253.  doi: 10.1070/SM1974v023n02ABEH001719.

[16]

R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys, 39 (1984), 85-131.  doi: 10.1070/RM1984v039n06ABEH003182.

[17] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC press, 1999. 
[18]

R. C. Robinson and R. F. Williams, Finite stability is not generic, in Dynamical Systems, (eds. M. M. Peixoto), Academic Press, New York, (1973), 451–462.

[19]

S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.

[20]

S. Smale, The $\Omega$-stability theorem, in Proc. Sympos. Pure Math. (eds. S.-S. Chern and S. Smale), AMS, (1970), 289–297.

[21]

R. F. Williams, The "DA" maps of Smale and structural stability, in Proc. Sympos. Pure Math. (eds. S.-S. Chern and S. Smale), AMS, (1970), 329–334.

show all references

References:
[1]

S. Kh. Aranson and V. Z. Grines, The topological classification of cascades on closed two-dimensional manifolds, Russian Math. Surveys, 45 (1990), 1-35.  doi: 10.1070/RM1990v045n01ABEH002322.

[2]

S. Kh. Aranson and V. Z. Grines, Cascades on surfaces, in Dynamical Systems IX (eds. D. V. Anosov), Springer, (1995), 141–175. doi: 10.1007/978-3-662-03172-8_3.

[3]

S. Kh. Aranson, R. V. Plykin, A. Yu. Zhirov and E. V. Zhuzhoma, Exact upper bounds for the number of one-dimensional basic sets of surface A-diffeomorphisms, Journal of Dynamical and Control Systems, 3 (1997), 1-18.  doi: 10.1007/BF02471759.

[4]

R. Bowen, Periodic points and measures for Axiom a diffeomorphisms, Transactions of the American Mathematical Society, 154 (1971), 377-397.  doi: 10.2307/1995452.

[5]

V. Z. Grines, The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. I(in Russian), Tr. Mosk. Mat. Obs., 32 (1975), 35-60. 

[6]

V. Z. Grines, On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers, Sb. Math., 188 (1997), 537-569.  doi: 10.1070/SM1997v188n04ABEH000216.

[7]

V. Z. Grines, On topological classification of A-diffeomorphisms of surfaces, Journal of Dynamical and Control Systems, 6 (2000), 97-126.  doi: 10.1023/A:1009573706584.

[8]

V. Z. Grines and Kh. Kh. Kalai, Diffeomorphisms of two-dimensional manifolds with spatially situated basic sets, Russian Uspekhi Math. Surveys, 40 (1985), 221-222. 

[9]

V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2-and 3-Manifolds, Springer, 2016. doi: 10.1007/978-3-319-44847-3.

[10]

V. Z. Grines, O. V. Pochinka and S. van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749.  doi: 10.17323/1609-4514-2016-16-4-727-749.

[11]

M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.

[12] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge university press, 1997. 
[13]

J. Palis, On the $C^1$ $\Omega$-stability conjecture, Publ. Math. IHES, 66 (1988), 211-215. 

[14]

R. V. Plykin, The topology of basis sets for Smale diffeomorphisms, Math. USSR-Sb., 13 (1971), 297-307.  doi: 10.1070/SM1971v013n02ABEH001026.

[15]

R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Math. USSR-Sb., 23 (1974), 233-253.  doi: 10.1070/SM1974v023n02ABEH001719.

[16]

R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys, 39 (1984), 85-131.  doi: 10.1070/RM1984v039n06ABEH003182.

[17] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC press, 1999. 
[18]

R. C. Robinson and R. F. Williams, Finite stability is not generic, in Dynamical Systems, (eds. M. M. Peixoto), Academic Press, New York, (1973), 451–462.

[19]

S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.

[20]

S. Smale, The $\Omega$-stability theorem, in Proc. Sympos. Pure Math. (eds. S.-S. Chern and S. Smale), AMS, (1970), 289–297.

[21]

R. F. Williams, The "DA" maps of Smale and structural stability, in Proc. Sympos. Pure Math. (eds. S.-S. Chern and S. Smale), AMS, (1970), 329–334.

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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