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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

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

[26]

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

[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. 

[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.
[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

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

[26]

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

[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. 

[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.
[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

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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