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November  2020, 0: 331-348. doi: 10.3934/jmd.2020012

Ergodicity and partial hyperbolicity on Seifert manifolds

Andy Hammerlindl 1, , Jana Rodriguez Hertz 2, and  Raúl Ures 2,

1. 

School of Mathematics, Monash University, Victoria 3800, Australia
2. 

Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, 1088 Xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong 518055, China

Received  July 19, 2019 Revised  September 17, 2020 Published  November 2020

Fund Project: This research was partially supported by the Australian Research Council. JRH: Partially supported by NSFC 11871262 and NSFC 11871394. RU: Partially supported by NSFC 11871262.

We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.

Keywords: Partially hyperbolic diffeomorphisms, stable ergodicity, accessibility.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37A25.
Citation: Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012
References:
[1]

A. Avila, S. Crovisier and A. Wilkinson, $C^1$ density of stable ergodicity, arXiv: 1709.04983. Google Scholar

[2]

T. Barthelmé, S. R. Fenley, S. Frankel and R. Potrie, Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds, preprint, arXiv: 1801.00214. Google Scholar

[3]

C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie, Anomolous partially hyperbolic diffeomorphisms III: Abundance and incoherence, preprint, arXiv: 1706.04962. Google Scholar

[4]

C. Bonatti, L. J. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.  doi: 10.1017/S1474748002000142.  Google Scholar

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C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.  doi: 10.1016/j.top.2004.10.009.  Google Scholar

[6] M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004.   Google Scholar
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M. I. Brin and J. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.   Google Scholar

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M. Brittenham, Essential laminations in Seifert-fibered spaces, Topology, 32 (1993), 61-85.  doi: 10.1016/0040-9383(93)90038-W.  Google Scholar

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M. Brunella, Expansive flows on Seifert manifolds and on torus bundles, Bol. Soc. Brasil Mat., 24 (1993), 89-104.  doi: 10.1007/BF01231697.  Google Scholar

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K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489.  doi: 10.4007/annals.2010.171.451.  Google Scholar

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P. D. Carrasco, F. R. Hertz, J. R. Hertz and R. Ures, Partially hyperbolic dynamics in dimension three, Ergodic Theory Dynam. Systems, 38 (2018), 2801-2837.  doi: 10.1017/etds.2016.142.  Google Scholar

[12] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.   Google Scholar
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S. R. Fenley, Regulating flows, topology of foliations and rigidity, Trans. Amer. Math. Soc., 357 (2005), 4957-5000.  doi: 10.1090/S0002-9947-05-03644-5.  Google Scholar

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S. R. Fenley, Rigidity of pseudo-Anosov flows transverse to $\Bbb{R}$-covered foliations, Comment. Math. Helv., 88 (2013), 643-676.  doi: 10.4171/CMH/299.  Google Scholar

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S. R. Fenley and R. Potrie, Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds, preprint, arXiv: 1809.02284. Google Scholar

[16]

J. Franks, Anosov diffeomorphisms, Amer. Math. Soc., 14 (1970) 61–93.  Google Scholar

[17]

S. Gan and Y. Shi, Rigidity of center Lyapunov exponents and $su$-integrability, Comment. Math. Helv., 95 (2020), 569-592.  doi: 10.4171/CMH/497.  Google Scholar

[18]

É. Ghys, Groups acting on the circle, Enseign. Math., 47 (2001), 329-407.   Google Scholar

[19]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math., 140 (1994), 295-329.  doi: 10.2307/2118602.  Google Scholar

[20]

A. Hammerlindl, Ergodic components of partially hyperbolic systems, Comment. Math. Helv., 92 (2017), 131-184.  doi: 10.4171/CMH/409.  Google Scholar

[21]

A. Hammerlindl and R. Potrie, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, J. Topol., 8 (2015), 842-870.  doi: 10.1112/jtopol/jtv009.  Google Scholar

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A. Hammerlindl, R. Potrie and M. Shannon, Seifert manifolds admitting partially hyperbolic diffeomorphisms, J. Mod. Dyn., 12 (2018), 193-222.  doi: 10.3934/jmd.2018008.  Google Scholar

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A. Hammerlindl and R. Ures, Ergodicity and partial hyperbolicity on the 3-torus, Commun. Contemp. Math., 16 (2014), 1350038, 22 pp. doi: 10.1142/S0219199713500387.  Google Scholar

[24]

A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1962), 367-397.   Google Scholar

[25]

G. Hector and U. Hirsch, Introduction to the geometry of foliations. Part B. Foliations of codimension one, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1983. doi: 10.1007/978-3-322-85619-7.  Google Scholar

[26]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, (French) Inst. Hautes études Sci. Publ. Math., 49 (1979), 5–233.  Google Scholar

[27]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[28]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 51 (1980), 137–173.  Google Scholar

[29] K. Mann, Rigidity and flexibility of group actions on the circle. Handbook of group actions. Vol. IV, Adv. Lect. Math. (ALM), 41, Int. Press, omerville, MA, 2018.   Google Scholar
[30]

P. Mendes, On Anosov diffeomorphisms on the plane, Proc. Amer. Math. Soc., 63 (1977), 231-235.  doi: 10.1090/S0002-9939-1977-0461585-X.  Google Scholar

[31]

J. Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv., 32 (1958), 215-223.  doi: 10.1007/BF02564579.  Google Scholar

[32]

F. R. Hertz, M. A. R. Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2 (2008), 187-208.  doi: 10.3934/jmd.2008.2.187.  Google Scholar

[33]

F. R. Hertz, M. A. R. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007,103–109.  Google Scholar

[34]

F. R. Hertz, M. A. R. Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[35]

F. R. Hertz, M. A. R. Hertz and R. Ures, Tori with hyperbolic dynamics in 3-manifolds, J. Mod. Dyn., 5 (2011), 185-202.  doi: 10.3934/jmd.2011.5.185.  Google Scholar

[36]

F. R. Hertz, J. R. Hertz and R. Ures, Center-unstable foliations do not have compact leaves, Math. Res. Lett., 23 (2016), 1819-1832.  doi: 10.4310/MRL.2016.v23.n6.a11.  Google Scholar

[37]

R. Saghin and J. Yang, personal communication., Google Scholar

[38]

P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc., 15 (1983), 401-487.  doi: 10.1112/blms/15.5.401.  Google Scholar

[39]

J. Zhang, Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds, preprint, arXiv: 1701.06176. Google Scholar

show all references

References:
[1]

A. Avila, S. Crovisier and A. Wilkinson, $C^1$ density of stable ergodicity, arXiv: 1709.04983. Google Scholar

[2]

T. Barthelmé, S. R. Fenley, S. Frankel and R. Potrie, Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds, preprint, arXiv: 1801.00214. Google Scholar

[3]

C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie, Anomolous partially hyperbolic diffeomorphisms III: Abundance and incoherence, preprint, arXiv: 1706.04962. Google Scholar

[4]

C. Bonatti, L. J. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.  doi: 10.1017/S1474748002000142.  Google Scholar

[5]

C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.  doi: 10.1016/j.top.2004.10.009.  Google Scholar

[6] M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004.   Google Scholar
[7]

M. I. Brin and J. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.   Google Scholar

[8]

M. Brittenham, Essential laminations in Seifert-fibered spaces, Topology, 32 (1993), 61-85.  doi: 10.1016/0040-9383(93)90038-W.  Google Scholar

[9]

M. Brunella, Expansive flows on Seifert manifolds and on torus bundles, Bol. Soc. Brasil Mat., 24 (1993), 89-104.  doi: 10.1007/BF01231697.  Google Scholar

[10]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[11]

P. D. Carrasco, F. R. Hertz, J. R. Hertz and R. Ures, Partially hyperbolic dynamics in dimension three, Ergodic Theory Dynam. Systems, 38 (2018), 2801-2837.  doi: 10.1017/etds.2016.142.  Google Scholar

[12] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.   Google Scholar
[13]

S. R. Fenley, Regulating flows, topology of foliations and rigidity, Trans. Amer. Math. Soc., 357 (2005), 4957-5000.  doi: 10.1090/S0002-9947-05-03644-5.  Google Scholar

[14]

S. R. Fenley, Rigidity of pseudo-Anosov flows transverse to $\Bbb{R}$-covered foliations, Comment. Math. Helv., 88 (2013), 643-676.  doi: 10.4171/CMH/299.  Google Scholar

[15]

S. R. Fenley and R. Potrie, Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds, preprint, arXiv: 1809.02284. Google Scholar

[16]

J. Franks, Anosov diffeomorphisms, Amer. Math. Soc., 14 (1970) 61–93.  Google Scholar

[17]

S. Gan and Y. Shi, Rigidity of center Lyapunov exponents and $su$-integrability, Comment. Math. Helv., 95 (2020), 569-592.  doi: 10.4171/CMH/497.  Google Scholar

[18]

É. Ghys, Groups acting on the circle, Enseign. Math., 47 (2001), 329-407.   Google Scholar

[19]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math., 140 (1994), 295-329.  doi: 10.2307/2118602.  Google Scholar

[20]

A. Hammerlindl, Ergodic components of partially hyperbolic systems, Comment. Math. Helv., 92 (2017), 131-184.  doi: 10.4171/CMH/409.  Google Scholar

[21]

A. Hammerlindl and R. Potrie, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, J. Topol., 8 (2015), 842-870.  doi: 10.1112/jtopol/jtv009.  Google Scholar

[22]

A. Hammerlindl, R. Potrie and M. Shannon, Seifert manifolds admitting partially hyperbolic diffeomorphisms, J. Mod. Dyn., 12 (2018), 193-222.  doi: 10.3934/jmd.2018008.  Google Scholar

[23]

A. Hammerlindl and R. Ures, Ergodicity and partial hyperbolicity on the 3-torus, Commun. Contemp. Math., 16 (2014), 1350038, 22 pp. doi: 10.1142/S0219199713500387.  Google Scholar

[24]

A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1962), 367-397.   Google Scholar

[25]

G. Hector and U. Hirsch, Introduction to the geometry of foliations. Part B. Foliations of codimension one, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1983. doi: 10.1007/978-3-322-85619-7.  Google Scholar

[26]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, (French) Inst. Hautes études Sci. Publ. Math., 49 (1979), 5–233.  Google Scholar

[27]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[28]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 51 (1980), 137–173.  Google Scholar

[29] K. Mann, Rigidity and flexibility of group actions on the circle. Handbook of group actions. Vol. IV, Adv. Lect. Math. (ALM), 41, Int. Press, omerville, MA, 2018.   Google Scholar
[30]

P. Mendes, On Anosov diffeomorphisms on the plane, Proc. Amer. Math. Soc., 63 (1977), 231-235.  doi: 10.1090/S0002-9939-1977-0461585-X.  Google Scholar

[31]

J. Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv., 32 (1958), 215-223.  doi: 10.1007/BF02564579.  Google Scholar

[32]

F. R. Hertz, M. A. R. Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2 (2008), 187-208.  doi: 10.3934/jmd.2008.2.187.  Google Scholar

[33]

F. R. Hertz, M. A. R. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007,103–109.  Google Scholar

[34]

F. R. Hertz, M. A. R. Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[35]

F. R. Hertz, M. A. R. Hertz and R. Ures, Tori with hyperbolic dynamics in 3-manifolds, J. Mod. Dyn., 5 (2011), 185-202.  doi: 10.3934/jmd.2011.5.185.  Google Scholar

[36]

F. R. Hertz, J. R. Hertz and R. Ures, Center-unstable foliations do not have compact leaves, Math. Res. Lett., 23 (2016), 1819-1832.  doi: 10.4310/MRL.2016.v23.n6.a11.  Google Scholar

[37]

R. Saghin and J. Yang, personal communication., Google Scholar

[38]

P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc., 15 (1983), 401-487.  doi: 10.1112/blms/15.5.401.  Google Scholar

[39]

J. Zhang, Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds, preprint, arXiv: 1701.06176. Google Scholar

Figure 1.  Center segments in the proof of Lemma 6.3
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