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On a class of singularly perturbed elliptic systems with asymptotic phase segregation
On decomposition of ambient surfaces admitting $ A $-diffeomorphisms with non-trivial attractors and repellers
HSE University, Bolshaya Pecherskaya 25/12, Nizhny Novgorod, Russia, 603155 |
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.
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] | |
[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] | |
[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. |


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