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

Ergodicity and partial hyperbolicity on Seifert manifolds

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.

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.

[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.

[3]

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

[4]

C. BonattiL. 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.

[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.

[6] M. BrinD. 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. 
[7]

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

[8]

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

[9]

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

[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.

[11]

P. D. CarrascoF. R. HertzJ. 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.

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

[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.

[15]

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

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[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. 
[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.

[31]

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

[32]

F. R. HertzM. 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.

[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.

[34]

F. R. HertzM. 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.

[35]

F. R. HertzM. 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.

[36]

F. R. HertzJ. 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.

[37]

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

[38]

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

[39]

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

show all references

References:
[1]

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

[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.

[3]

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

[4]

C. BonattiL. 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.

[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.

[6] M. BrinD. 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. 
[7]

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

[8]

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

[9]

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

[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.

[11]

P. D. CarrascoF. R. HertzJ. 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.

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

[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.

[15]

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

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[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. 
[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.

[31]

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

[32]

F. R. HertzM. 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.

[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.

[34]

F. R. HertzM. 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.

[35]

F. R. HertzM. 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.

[36]

F. R. HertzJ. 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.

[37]

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

[38]

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

[39]

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

Figure 1.  Center segments in the proof of Lemma 6.3
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