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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.

[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.

[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]

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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[31]

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

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.

[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.

[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]

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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[31]

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

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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.

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