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March  2020, 40(3): 1879-1887. doi: 10.3934/dcds.2020097

Minimality and stable Bernoulliness in dimension 3

Gabriel Núñez 1,2, and  Jana Rodriguez Hertz 3,4,

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

Received  July 2019 Revised  September 2019 Published  December 2019

Fund Project: GN was supported by Agencia Nacional de Investigación e Innovación. The research that gives rise to the results presented in this publication received funds from the Agencia Nacional de Investigación e Innovación under the code POS_NAC_2014_1_102348
JRH was supported by SUSTech, the project NSFC 11871262 and the SUSTech International Center for Mathematics
It is NSFC No. 11871394 for JRH.

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.

Keywords: Stable ergodicity, stable minimality, smooth ergodic measures, generic properties, non-uniform hyperbolicity.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Gabriel Núñez, Jana Rodriguez Hertz. Minimality and stable Bernoulliness in dimension 3. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1879-1887. doi: 10.3934/dcds.2020097
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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