2021, 17: 557-584. doi: 10.3934/jmd.2021019

Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China

Received  November 14, 2017 Revised  May 16, 2021 Published  November 2021

We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Möbius band or a plane.

Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time $ m $-map (for $ m>0 $ large) of a non-transitive Anosov flow $ \phi_t $ on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to $ \phi_t $. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.

Citation: Jinhua Zhang. Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds. Journal of Modern Dynamics, 2021, 17: 557-584. doi: 10.3934/jmd.2021019
References:
[1]

D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, J. Mod. Dyn., 7 (2013), 565-604.  doi: 10.3934/jmd.2013.7.565.  Google Scholar

[2]

C. BonattiS. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 185-244.  doi: 10.1007/s10240-009-0021-z.  Google Scholar

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C. BonattiS. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1473-1508.  doi: 10.1017/S0143385707000090.  Google Scholar

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C. BonattiA. Gogolev and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅱ: Stably ergodic examples, Invent. Math., 206 (2016), 801-836.  doi: 10.1007/s00222-016-0663-7.  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

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C. Bonatti and J. Zhang, Transverse foliations on the torus ${\mathbb{T}}^2$ and partially hyperbolic diffeomorphisms on 3-manifolds, Comment. Math. Helv., 92 (2017), 513-550.  doi: 10.4171/CMH/418.  Google Scholar

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C. Bonatti and J. Zhang, Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center, Sci. China Math., 63 (2020), 1647-1670.  doi: 10.1007/s11425-019-1751-2.  Google Scholar

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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, 307–312.  Google Scholar

[10]

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

[11]

M. Brunella, Separating the basic sets of a nontransitive Anosov flow, Bull. London Math. Soc., 25 (1993), 487-490.  doi: 10.1112/blms/25.5.487.  Google Scholar

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C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5292-4.  Google Scholar

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A. Candel and L. Conlon, Foliations. II, Graduate Studies in Mathematics, 60, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/060.  Google Scholar

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P. D. Carrasco, Compact dynamical foliations, Ergodic Theory Dynam. Systems, 35 (2015), 2474-2498.  doi: 10.1017/etds.2014.42.  Google Scholar

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

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S. Crovisier and R. Potrie, Introduction to Partially Hyperbolic Dynamics, Lecture notes for a minicourse at ICTP (2015). Available on the webpage of the conference and the authors. Google Scholar

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B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds, I: Actions of nonlinear groups, What's Next? The Mathematical Legacy of William P. Thurston, Ann. of Math. Stud., 205, Princeton Univ. Press, Princeton, NJ, 116–140. doi: 10.2307/j.ctvthhdvv.9.  Google Scholar

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J. Franks and B. Williams, Anomalous Anosov flows, Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, 1980,158–174.  Google Scholar

[19]

A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations, J. Mod. Dyn., 5 (2011), 747-769.  doi: 10.3934/jmd.2011.5.747.  Google Scholar

[20]

H. B. Griffiths, The fundamental group of a surface, and a theorem of Schreier, Acta Math., 110 (1963), 1-17.  doi: 10.1007/BF02391853.  Google Scholar

[21]

A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. Lond. Math. Soc. (2), 89 (2014), 853-875.  doi: 10.1112/jlms/jdu013.  Google Scholar

[22]

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

[23]

A. Hammerlindl and R. Potrie, Partial hyperbolicity and classification: A survey, Ergodic Theory Dynam. Systems, 38 (2018), 401-443.  doi: 10.1017/etds.2016.50.  Google Scholar

[24] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.   Google Scholar
[25]

M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152.  doi: 10.1090/S0002-9904-1969-12184-1.  Google Scholar

[26]

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

[27]

O. Hölder, Die Axiome der Quantität und die Lehre vom Mass, Ber. Verh. Sachs. Ges. Wiss. Leipzig, Math. Phys. C1., 53 (1901), 1-64.   Google Scholar

[28] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[29]

S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč, 14 (1965), 248–278.  Google Scholar

[30]

C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity, 13 (1997), 125-179.  doi: 10.1006/jcom.1997.0437.  Google Scholar

[31]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Partially Hyperbolic Dynamics, Laminations, and Teichmüler Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007, 35–87.  Google Scholar

[32]

F. Rodriguez Hertz, M. A. Rodriguez 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

[33]

F. Rodriguez HertzM. A. Rodriguez 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

[34]

F. Rodriguez HertzJ. Rodriguez Hertz and R. Ures, A non-dynamically coherent example on ${\mathbb{T}}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.  Google Scholar

[35]

F. Rodriguez HertzJ. Rodriguez 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

[36]

F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures, Partial Hyperbolicity in 3-Manifolds, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011.  Google Scholar

[37]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

show all references

References:
[1]

D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, J. Mod. Dyn., 7 (2013), 565-604.  doi: 10.3934/jmd.2013.7.565.  Google Scholar

[2]

C. BonattiS. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 185-244.  doi: 10.1007/s10240-009-0021-z.  Google Scholar

[3]

C. Bonatti, K. Parwani and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅰ: Dynamically coherent examples, Ann. Sci. Éc. Norm. Supér (4), 49, (2016) 1387–1402. doi: 10.24033/asens.2311.  Google Scholar

[4]

C. BonattiS. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1473-1508.  doi: 10.1017/S0143385707000090.  Google Scholar

[5]

C. BonattiA. Gogolev and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅱ: Stably ergodic examples, Invent. Math., 206 (2016), 801-836.  doi: 10.1007/s00222-016-0663-7.  Google Scholar

[6]

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

[7]

C. Bonatti and J. Zhang, Transverse foliations on the torus ${\mathbb{T}}^2$ and partially hyperbolic diffeomorphisms on 3-manifolds, Comment. Math. Helv., 92 (2017), 513-550.  doi: 10.4171/CMH/418.  Google Scholar

[8]

C. Bonatti and J. Zhang, Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center, Sci. China Math., 63 (2020), 1647-1670.  doi: 10.1007/s11425-019-1751-2.  Google Scholar

[9]

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, 307–312.  Google Scholar

[10]

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

[11]

M. Brunella, Separating the basic sets of a nontransitive Anosov flow, Bull. London Math. Soc., 25 (1993), 487-490.  doi: 10.1112/blms/25.5.487.  Google Scholar

[12]

C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5292-4.  Google Scholar

[13]

A. Candel and L. Conlon, Foliations. II, Graduate Studies in Mathematics, 60, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/060.  Google Scholar

[14]

P. D. Carrasco, Compact dynamical foliations, Ergodic Theory Dynam. Systems, 35 (2015), 2474-2498.  doi: 10.1017/etds.2014.42.  Google Scholar

[15]

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

[16]

S. Crovisier and R. Potrie, Introduction to Partially Hyperbolic Dynamics, Lecture notes for a minicourse at ICTP (2015). Available on the webpage of the conference and the authors. Google Scholar

[17]

B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds, I: Actions of nonlinear groups, What's Next? The Mathematical Legacy of William P. Thurston, Ann. of Math. Stud., 205, Princeton Univ. Press, Princeton, NJ, 116–140. doi: 10.2307/j.ctvthhdvv.9.  Google Scholar

[18]

J. Franks and B. Williams, Anomalous Anosov flows, Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, 1980,158–174.  Google Scholar

[19]

A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations, J. Mod. Dyn., 5 (2011), 747-769.  doi: 10.3934/jmd.2011.5.747.  Google Scholar

[20]

H. B. Griffiths, The fundamental group of a surface, and a theorem of Schreier, Acta Math., 110 (1963), 1-17.  doi: 10.1007/BF02391853.  Google Scholar

[21]

A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. Lond. Math. Soc. (2), 89 (2014), 853-875.  doi: 10.1112/jlms/jdu013.  Google Scholar

[22]

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

[23]

A. Hammerlindl and R. Potrie, Partial hyperbolicity and classification: A survey, Ergodic Theory Dynam. Systems, 38 (2018), 401-443.  doi: 10.1017/etds.2016.50.  Google Scholar

[24] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.   Google Scholar
[25]

M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152.  doi: 10.1090/S0002-9904-1969-12184-1.  Google Scholar

[26]

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

[27]

O. Hölder, Die Axiome der Quantität und die Lehre vom Mass, Ber. Verh. Sachs. Ges. Wiss. Leipzig, Math. Phys. C1., 53 (1901), 1-64.   Google Scholar

[28] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[29]

S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč, 14 (1965), 248–278.  Google Scholar

[30]

C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity, 13 (1997), 125-179.  doi: 10.1006/jcom.1997.0437.  Google Scholar

[31]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Partially Hyperbolic Dynamics, Laminations, and Teichmüler Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007, 35–87.  Google Scholar

[32]

F. Rodriguez Hertz, M. A. Rodriguez 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

[33]

F. Rodriguez HertzM. A. Rodriguez 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

[34]

F. Rodriguez HertzJ. Rodriguez Hertz and R. Ures, A non-dynamically coherent example on ${\mathbb{T}}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.  Google Scholar

[35]

F. Rodriguez HertzJ. Rodriguez 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

[36]

F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures, Partial Hyperbolicity in 3-Manifolds, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011.  Google Scholar

[37]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

Figure 1.  The intersection between the local stable manifold of $ f^{n_i}(L) $ and the local unstable manifold of $ \mathscr{F}^c(w_1) $
Figure 2.  Defining $ h^s $ on a center stable leaf restricted to $ \tilde{M}^-\backslash\tilde{\mathscr{A}} $
Figure 3.  The real lines and the dashed lines denote the leaves of the lifts of foliations $ {\mathscr{F}}^s_i $ and $ {\mathscr{F}}^u_i $ to the universal cover, respectively
Figure 4.  The real lines and the dashed lines denote the leaves of the lifts of foliations $ {\mathscr{F}}^u_i $ and $ {\mathscr{F}}^s_i $ to the universal cover, respectively
Figure 5.  The dashed line denotes the center leaf obtained by the intersection of center stable and center unstable leaves
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