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Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition
Minimality and stable Bernoulliness in dimension 3
1. | IMERL, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, 11.300 Montevideo, Uruguay |
2. | Departamento de Ciencias Exactas y Naturales, Facultad de Ingeniería y Tecnologías, Universidad Católica del Uruguay, Comandante Braga 2715, 11.600 Montevideo, Uruguay |
3. | Department of Mathematics, Southern University of Science and Technology of China, No 1088, Xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong 518055, China |
4. | SUSTech International Center for Mathematics, No 1088, Xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong 518055, China |
In 3-dimensional manifolds, we prove that generically in $ \operatorname{Diff}^{1}_{m}(M^{3}) $, the existence of a minimal expanding invariant foliation implies stable Bernoulliness.
References:
[1] |
F. Abdenur, C. Bonatti and S. Crovisier,
Nonuniform hyperbolicity for $\mathcal{C}^1$-generic diffeomorphisms, Israel Journal of Mathematics, 183 (2011), 1-60.
doi: 10.1007/s11856-011-0041-5. |
[2] |
F. Abdenur and S. Crovisier, Transitivity and topological mixing for $C^1$ diffeomorphisms, Essays in Mathematics and its Applications, Springer, Heidelberg, (2012), 1–16.
doi: 10.1007/978-3-642-28821-0_1. |
[3] |
D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Ccurvature, American Mathematical Society, Providence, R.I. 1969 |
[4] |
D. V. Anosov and Y. G. Sinai,
Some smooth ergodic systems, Uspehi Mat. Nauk, 22 (1967), 107-172.
|
[5] |
A. Avila and J. Bochi,
Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Transactions of the American Mathematical Society, 364 (2012), 2883-2907.
doi: 10.1090/S0002-9947-2012-05423-7. |
[6] |
A. Avila, S. Crovisier and A. Wilkinson,
Diffeomorphisms with positive metric entropy, Publ. Math. Inst. Hautes Ëtudes Sci., 124 (2016), 319-347.
doi: 10.1007/s10240-016-0086-4. |
[7] |
A. Avila, S. Crovisier and A. Wilkinson, $C^1$ density of stable ergodicity, (2017). |
[8] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[9] |
C. Bonatti and S. Crovisier,
Récurrence et généricité, Inventiones Mathematicae, 158 (2004), 33-104.
doi: 10.1007/s00222-004-0368-1. |
[10] |
C. Bonatti, L. J. Díaz and E. R. Pujals,
A c1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Mathematics, 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[11] |
C. Bonatti and M. Viana,
SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics, 115 (2000), 157-193.
doi: 10.1007/BF02810585. |
[12] |
L. J. Díaz, E. R. Pujals and R. Ures,
Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.
doi: 10.1007/BF02392945. |
[13] |
M. Grayson, C. Pugh and M. Shub,
Stably ergodic diffeomorphisms, The Annals of Mathematics, 140 (1994), 295-329.
doi: 10.2307/2118602. |
[14] |
E. Hopf,
Statistik der geod tischen linien in mannigfaltigkeiten negativer krümmung, Berichten der S chsischen Akademie der Wissenschaften zu Leipzi, 91 (1939), 261-304.
|
[15] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Hautes Études Sci. Publ. Math., (1980), 137–173. |
[16] |
R. Mañé,
Oseledec's theorem from the generic viewpoint, Proceedings of the International Congress of Mathematicians, PWN, Warsaw, 1/2 (1984), 1269-1276.
|
[17] |
D. Obata,
On the stable ergodicity of diffeomorphisms with dominated splitting, Nonlinearity, 32 (2019), 445-463.
doi: 10.1088/1361-6544/aaea93. |
[18] |
J. B. Pesin,
Characteristic Lyapunov exponents and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-114.
|
[19] |
C. Pugh and M. Shub,
Stable ergodicity and partial hyperbolicity, International Conference on Dynamical Systems, Pitman Res. Notes Math. Ser., Longman, Harlow, 362 (1996), 182-187.
|
[20] |
C. C. Pugh, The ${C^{1+\alpha}}$ hypothesis in Pesin theory, Inst. Hautes Études Sci. Publ. Math., (1984), 143–161. |
[21] |
J. Rodriguez Hertz,
Genericity of nonuniform hyperbolicity in dimension 3, Journal of Modern Dynamics, 6 (2012), 121-138.
doi: 10.3934/jmd.2012.6.121. |
[22] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures,
New criteria for ergodicity and nonuniform hyperbolicity, Duke Mathematical Journal, 160 (2011), 599-629.
doi: 10.1215/00127094-1444314. |
[23] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,
Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Inventiones Mathematicae, 172 (2008), 353-381.
doi: 10.1007/s00222-007-0100-z. |
[24] |
A. Tahzibi,
Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel Journal of Mathematics, 142 (2004), 315-344.
doi: 10.1007/BF02771539. |
[25] |
A. Wilkinson,
Stable ergodicity of the time-one map of a geodesic flow, Ergodic Theory and Dynamical Systems, 18 (1998), 1545-1587.
doi: 10.1017/S0143385798117984. |
show all references
References:
[1] |
F. Abdenur, C. Bonatti and S. Crovisier,
Nonuniform hyperbolicity for $\mathcal{C}^1$-generic diffeomorphisms, Israel Journal of Mathematics, 183 (2011), 1-60.
doi: 10.1007/s11856-011-0041-5. |
[2] |
F. Abdenur and S. Crovisier, Transitivity and topological mixing for $C^1$ diffeomorphisms, Essays in Mathematics and its Applications, Springer, Heidelberg, (2012), 1–16.
doi: 10.1007/978-3-642-28821-0_1. |
[3] |
D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Ccurvature, American Mathematical Society, Providence, R.I. 1969 |
[4] |
D. V. Anosov and Y. G. Sinai,
Some smooth ergodic systems, Uspehi Mat. Nauk, 22 (1967), 107-172.
|
[5] |
A. Avila and J. Bochi,
Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Transactions of the American Mathematical Society, 364 (2012), 2883-2907.
doi: 10.1090/S0002-9947-2012-05423-7. |
[6] |
A. Avila, S. Crovisier and A. Wilkinson,
Diffeomorphisms with positive metric entropy, Publ. Math. Inst. Hautes Ëtudes Sci., 124 (2016), 319-347.
doi: 10.1007/s10240-016-0086-4. |
[7] |
A. Avila, S. Crovisier and A. Wilkinson, $C^1$ density of stable ergodicity, (2017). |
[8] |
J. Bochi,
Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[9] |
C. Bonatti and S. Crovisier,
Récurrence et généricité, Inventiones Mathematicae, 158 (2004), 33-104.
doi: 10.1007/s00222-004-0368-1. |
[10] |
C. Bonatti, L. J. Díaz and E. R. Pujals,
A c1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Mathematics, 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[11] |
C. Bonatti and M. Viana,
SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics, 115 (2000), 157-193.
doi: 10.1007/BF02810585. |
[12] |
L. J. Díaz, E. R. Pujals and R. Ures,
Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.
doi: 10.1007/BF02392945. |
[13] |
M. Grayson, C. Pugh and M. Shub,
Stably ergodic diffeomorphisms, The Annals of Mathematics, 140 (1994), 295-329.
doi: 10.2307/2118602. |
[14] |
E. Hopf,
Statistik der geod tischen linien in mannigfaltigkeiten negativer krümmung, Berichten der S chsischen Akademie der Wissenschaften zu Leipzi, 91 (1939), 261-304.
|
[15] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Hautes Études Sci. Publ. Math., (1980), 137–173. |
[16] |
R. Mañé,
Oseledec's theorem from the generic viewpoint, Proceedings of the International Congress of Mathematicians, PWN, Warsaw, 1/2 (1984), 1269-1276.
|
[17] |
D. Obata,
On the stable ergodicity of diffeomorphisms with dominated splitting, Nonlinearity, 32 (2019), 445-463.
doi: 10.1088/1361-6544/aaea93. |
[18] |
J. B. Pesin,
Characteristic Lyapunov exponents and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-114.
|
[19] |
C. Pugh and M. Shub,
Stable ergodicity and partial hyperbolicity, International Conference on Dynamical Systems, Pitman Res. Notes Math. Ser., Longman, Harlow, 362 (1996), 182-187.
|
[20] |
C. C. Pugh, The ${C^{1+\alpha}}$ hypothesis in Pesin theory, Inst. Hautes Études Sci. Publ. Math., (1984), 143–161. |
[21] |
J. Rodriguez Hertz,
Genericity of nonuniform hyperbolicity in dimension 3, Journal of Modern Dynamics, 6 (2012), 121-138.
doi: 10.3934/jmd.2012.6.121. |
[22] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures,
New criteria for ergodicity and nonuniform hyperbolicity, Duke Mathematical Journal, 160 (2011), 599-629.
doi: 10.1215/00127094-1444314. |
[23] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,
Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Inventiones Mathematicae, 172 (2008), 353-381.
doi: 10.1007/s00222-007-0100-z. |
[24] |
A. Tahzibi,
Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel Journal of Mathematics, 142 (2004), 315-344.
doi: 10.1007/BF02771539. |
[25] |
A. Wilkinson,
Stable ergodicity of the time-one map of a geodesic flow, Ergodic Theory and Dynamical Systems, 18 (1998), 1545-1587.
doi: 10.1017/S0143385798117984. |
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