October  2013, 7(4): 565-604. doi: 10.3934/jmd.2013.7.565

Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation

1. 

Institut de Mathématiques de Bourgogne CNRS - URM 5584 Université de Bourgogne Dijon 21004, France

Received  March 2013 Revised  November 2013 Published  March 2014

We consider a partially hyperbolic $C^1$-diffeomorphism $f\colon M \rightarrow M$ with a uniformly compact $f$-invariant center foliation $\mathcal{F}^c$. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space $M/\mathcal{F}^c$ of the center foliation is a torus and $f$ induces a hyperbolic automorphism on it, in particular, $f$ is centrally transitive.
    We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy-invariant family of measures on the unstable leaves (Margulis measure).
Citation: Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565
References:
[1]

D. Bohnet and C. Bonatti, Partially hyperbolic diffeomorphism with uniformly compact center foliation: The quotient dynamics, arXiv:1210.2835, 2012. Google Scholar

[2]

R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2), 56 (1952), 354-362. doi: 10.2307/1969804.  Google Scholar

[3]

S. Bochner, Compact groups of differentiable transformations, Ann. of Math. (2), 46 (1945), 372-381. doi: 10.2307/1969157.  Google Scholar

[4]

D. Bohnet, Partially Hyperbolic Diffeomorphisms with a Compact Center Foliation with Finite Holonomy, Ph.D thesis, University of Hamburg, 2011. Google Scholar

[5]

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

[6]

L. E. J. Brouwer, Über die periodischen Transformationen der Kugel, Mathematische Annalen, 80 (1919), 39-41. doi: 10.1007/BF01463233.  Google Scholar

[7]

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

[8]

P. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.  Google Scholar

[9]

A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[10]

H. Colman and S. Hurder, Ls-category of compact Hausdorff foliations, Trans. Amer. Math. Soc., 356 (2004), 1463-1487. doi: 10.1090/S0002-9947-03-03459-7.  Google Scholar

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A. Constantin and B. Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. (2), 40 (1994), 193-204.  Google Scholar

[12]

C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.  Google Scholar

[13]

S. Eilenberg, Sur les transformations périodiques de la surface de sphère, Fund. Math., 22 (1934), 28-41. Google Scholar

[14]

D. B. A. Epstein, K. Millet and D. Tischler, Leaves without holonomy, J. London Math. Soc. (2), 16 (1977), 548-552.  Google Scholar

[15]

D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2), 95 (1972), 66-82. doi: 10.2307/1970854.  Google Scholar

[16]

D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265-282. doi: 10.5802/aif.607.  Google Scholar

[17]

D. B. A. Epstein and E. Vogt, A counterexample to the periodic orbit conjecture in codimension 3, Ann. of Math. (2), 108 (1978), 539-552. doi: 10.2307/1971187.  Google Scholar

[18]

J. Franks, Anosov diffeomorphisms, in 1970 Global Analysis (Proceedings of Symposia in Pure Mathematics, Vol. XIV, Berkeley, Calif., 1968), AMS, Providence, RI, 1970, 61-93.  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]

G. Hector, Feuilletages en cylindres, in Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, 252-270.  Google Scholar

[21]

K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 801-806. doi: 10.1017/S0143385701001390.  Google Scholar

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B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[23]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

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F. Rodriguez Hertz, M. A. Rodriguez-Hertz and R. Ures, A non-dynamically coherent example in 3-torus, preprint, 2010. Google Scholar

[25]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Math. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.  Google Scholar

[26]

G. A. Margulis, Certain measures that are connected with u-flows on compact manifolds, Functional Anal. Appl., 4 (1970), 55-67.  Google Scholar

[27]

K. C. Millett, Compact foliations, in Differential Topology and Geometry (Proc. Colloq., Dijon, 1974), Lecture Notes in Math., Vol. 484, Springer, Berlin, 1975, 277-287.  Google Scholar

[28]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar

[29]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 42 (1941), 874-920. doi: 10.2307/1968772.  Google Scholar

[30]

R. Potrie and A. Hammerlindl, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, arXiv:1307.4631, 2013. Google Scholar

[31]

C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math., 15 (1972), 1-23. doi: 10.1007/BF01418639.  Google Scholar

[32]

G. Reeb, Sur Certaines Propriétés Topologiques des Variétés Feuilletées, Publ. Inst. Math. Univ. Strasbourg 11, Actualités Sci. Ind., no. 1183, Hermann & Cie., Paris, 1952, 5-89, 155-156.  Google Scholar

[33]

Walter Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[34]

D. Sullivan, A counterexample to the periodic orbit conjecture, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 5-14.  Google Scholar

[35]

D. Sullivan, A new flow, Bull. Amer. Math. Sco., 82 (1976), 331-332. doi: 10.1090/S0002-9904-1976-14047-5.  Google Scholar

[36]

J. Vieitez, A 3D-manifold with a uniform local product structure is T3, Publ. Mat. Urug., 8 (1999), 47-62.  Google Scholar

[37]

B. von Kerékjártó, Über Transformationen des ebenen Kreisringes, Mathematische Annalen, 80 (1919), 33-35. Google Scholar

[38]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergodic Theory Dynam. Systems, 18 (1998), 1545-1587. doi: 10.1017/S0143385798117984.  Google Scholar

show all references

References:
[1]

D. Bohnet and C. Bonatti, Partially hyperbolic diffeomorphism with uniformly compact center foliation: The quotient dynamics, arXiv:1210.2835, 2012. Google Scholar

[2]

R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2), 56 (1952), 354-362. doi: 10.2307/1969804.  Google Scholar

[3]

S. Bochner, Compact groups of differentiable transformations, Ann. of Math. (2), 46 (1945), 372-381. doi: 10.2307/1969157.  Google Scholar

[4]

D. Bohnet, Partially Hyperbolic Diffeomorphisms with a Compact Center Foliation with Finite Holonomy, Ph.D thesis, University of Hamburg, 2011. Google Scholar

[5]

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

[6]

L. E. J. Brouwer, Über die periodischen Transformationen der Kugel, Mathematische Annalen, 80 (1919), 39-41. doi: 10.1007/BF01463233.  Google Scholar

[7]

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

[8]

P. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.  Google Scholar

[9]

A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[10]

H. Colman and S. Hurder, Ls-category of compact Hausdorff foliations, Trans. Amer. Math. Soc., 356 (2004), 1463-1487. doi: 10.1090/S0002-9947-03-03459-7.  Google Scholar

[11]

A. Constantin and B. Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. (2), 40 (1994), 193-204.  Google Scholar

[12]

C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.  Google Scholar

[13]

S. Eilenberg, Sur les transformations périodiques de la surface de sphère, Fund. Math., 22 (1934), 28-41. Google Scholar

[14]

D. B. A. Epstein, K. Millet and D. Tischler, Leaves without holonomy, J. London Math. Soc. (2), 16 (1977), 548-552.  Google Scholar

[15]

D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2), 95 (1972), 66-82. doi: 10.2307/1970854.  Google Scholar

[16]

D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265-282. doi: 10.5802/aif.607.  Google Scholar

[17]

D. B. A. Epstein and E. Vogt, A counterexample to the periodic orbit conjecture in codimension 3, Ann. of Math. (2), 108 (1978), 539-552. doi: 10.2307/1971187.  Google Scholar

[18]

J. Franks, Anosov diffeomorphisms, in 1970 Global Analysis (Proceedings of Symposia in Pure Mathematics, Vol. XIV, Berkeley, Calif., 1968), AMS, Providence, RI, 1970, 61-93.  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]

G. Hector, Feuilletages en cylindres, in Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, 252-270.  Google Scholar

[21]

K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 801-806. doi: 10.1017/S0143385701001390.  Google Scholar

[22]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[23]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[24]

F. Rodriguez Hertz, M. A. Rodriguez-Hertz and R. Ures, A non-dynamically coherent example in 3-torus, preprint, 2010. Google Scholar

[25]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Math. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.  Google Scholar

[26]

G. A. Margulis, Certain measures that are connected with u-flows on compact manifolds, Functional Anal. Appl., 4 (1970), 55-67.  Google Scholar

[27]

K. C. Millett, Compact foliations, in Differential Topology and Geometry (Proc. Colloq., Dijon, 1974), Lecture Notes in Math., Vol. 484, Springer, Berlin, 1975, 277-287.  Google Scholar

[28]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar

[29]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 42 (1941), 874-920. doi: 10.2307/1968772.  Google Scholar

[30]

R. Potrie and A. Hammerlindl, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, arXiv:1307.4631, 2013. Google Scholar

[31]

C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math., 15 (1972), 1-23. doi: 10.1007/BF01418639.  Google Scholar

[32]

G. Reeb, Sur Certaines Propriétés Topologiques des Variétés Feuilletées, Publ. Inst. Math. Univ. Strasbourg 11, Actualités Sci. Ind., no. 1183, Hermann & Cie., Paris, 1952, 5-89, 155-156.  Google Scholar

[33]

Walter Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[34]

D. Sullivan, A counterexample to the periodic orbit conjecture, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 5-14.  Google Scholar

[35]

D. Sullivan, A new flow, Bull. Amer. Math. Sco., 82 (1976), 331-332. doi: 10.1090/S0002-9904-1976-14047-5.  Google Scholar

[36]

J. Vieitez, A 3D-manifold with a uniform local product structure is T3, Publ. Mat. Urug., 8 (1999), 47-62.  Google Scholar

[37]

B. von Kerékjártó, Über Transformationen des ebenen Kreisringes, Mathematische Annalen, 80 (1919), 33-35. Google Scholar

[38]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergodic Theory Dynam. Systems, 18 (1998), 1545-1587. doi: 10.1017/S0143385798117984.  Google Scholar

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