# American Institue of Mathematical Sciences

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,, , (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. doi: 10.2307/1969804. [3] S. Bochner, Compact groups of differentiable transformations,, Ann. of Math. (2), 46 (1945), 372. doi: 10.2307/1969157. [4] D. Bohnet, Partially Hyperbolic Diffeomorphisms with a Compact Center Foliation with Finite Holonomy,, Ph.D thesis, (2011). [5] M. I. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170. [6] L. E. J. Brouwer, Über die periodischen Transformationen der Kugel,, Mathematische Annalen, 80 (1919), 39. doi: 10.1007/BF01463233. [7] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds,, Topology, 44 (2005), 475. doi: 10.1016/j.top.2004.10.009. [8] P. Carrasco, Compact Dynamical Foliations,, Ph.D thesis, (2011). [9] A. Candel and L. Conlon, Foliations. I,, Graduate Studies in Mathematics, (2000). [10] H. Colman and S. Hurder, Ls-category of compact Hausdorff foliations,, Trans. Amer. Math. Soc., 356 (2004), 1463. 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. [12] C. Conley, Isolated invariant sets and the Morse index,, CBMS Regional Conference Series in Mathematics, (1978). [13] S. Eilenberg, Sur les transformations périodiques de la surface de sphère,, Fund. Math., 22 (1934), 28. [14] D. B. A. Epstein, K. Millet and D. Tischler, Leaves without holonomy,, J. London Math. Soc. (2), 16 (1977), 548. [15] D. B. A. Epstein, Periodic flows on three-manifolds,, Ann. of Math. (2), 95 (1972), 66. doi: 10.2307/1970854. [16] D. B. A. Epstein, Foliations with all leaves compact,, Ann. Inst. Fourier (Grenoble), 26 (1976), 265. 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. doi: 10.2307/1971187. [18] J. Franks, Anosov diffeomorphisms,, in 1970 Global Analysis (Proceedings of Symposia in Pure Mathematics, (1968), 61. [19] A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations,, J. Mod. Dyn., 5 (2011), 747. doi: 10.3934/jmd.2011.5.747. [20] G. Hector, Feuilletages en cylindres,, in Geometry and Topology (Proc. III Latin Amer. School of Math., (1976), 252. [21] K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 21 (2001), 801. doi: 10.1017/S0143385701001390. [22] B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). [23] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds,, Bull. Amer. Math. Soc., 76 (1970), 1015. 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. 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. [27] K. C. Millett, Compact foliations,, in Differential Topology and Geometry (Proc. Colloq., (1974), 277. [28] S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761. 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. doi: 10.2307/1968772. [30] R. Potrie and A. Hammerlindl, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group,, , (2013). [31] C. Pugh and M. Shub, Ergodicity of Anosov actions,, Invent. Math., 15 (1972), 1. doi: 10.1007/BF01418639. [32] G. Reeb, Sur Certaines Propriétés Topologiques des Variétés Feuilletées,, Publ. Inst. Math. Univ. Strasbourg 11, (1183), 5. [33] Walter Rudin, Real and Complex Analysis,, Third edition, (1987). [34] D. Sullivan, A counterexample to the periodic orbit conjecture,, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 5. [35] D. Sullivan, A new flow,, Bull. Amer. Math. Sco., 82 (1976), 331. 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. [37] B. von Kerékjártó, Über Transformationen des ebenen Kreisringes,, Mathematische Annalen, 80 (1919), 33. [38] A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow,, Ergodic Theory Dynam. Systems, 18 (1998), 1545. doi: 10.1017/S0143385798117984.

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##### References:
 [1] D. Bohnet and C. Bonatti, Partially hyperbolic diffeomorphism with uniformly compact center foliation: The quotient dynamics,, , (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. doi: 10.2307/1969804. [3] S. Bochner, Compact groups of differentiable transformations,, Ann. of Math. (2), 46 (1945), 372. doi: 10.2307/1969157. [4] D. Bohnet, Partially Hyperbolic Diffeomorphisms with a Compact Center Foliation with Finite Holonomy,, Ph.D thesis, (2011). [5] M. I. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170. [6] L. E. J. Brouwer, Über die periodischen Transformationen der Kugel,, Mathematische Annalen, 80 (1919), 39. doi: 10.1007/BF01463233. [7] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds,, Topology, 44 (2005), 475. doi: 10.1016/j.top.2004.10.009. [8] P. Carrasco, Compact Dynamical Foliations,, Ph.D thesis, (2011). [9] A. Candel and L. Conlon, Foliations. I,, Graduate Studies in Mathematics, (2000). [10] H. Colman and S. Hurder, Ls-category of compact Hausdorff foliations,, Trans. Amer. Math. Soc., 356 (2004), 1463. 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. [12] C. Conley, Isolated invariant sets and the Morse index,, CBMS Regional Conference Series in Mathematics, (1978). [13] S. Eilenberg, Sur les transformations périodiques de la surface de sphère,, Fund. Math., 22 (1934), 28. [14] D. B. A. Epstein, K. Millet and D. Tischler, Leaves without holonomy,, J. London Math. Soc. (2), 16 (1977), 548. [15] D. B. A. Epstein, Periodic flows on three-manifolds,, Ann. of Math. (2), 95 (1972), 66. doi: 10.2307/1970854. [16] D. B. A. Epstein, Foliations with all leaves compact,, Ann. Inst. Fourier (Grenoble), 26 (1976), 265. 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. doi: 10.2307/1971187. [18] J. Franks, Anosov diffeomorphisms,, in 1970 Global Analysis (Proceedings of Symposia in Pure Mathematics, (1968), 61. [19] A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations,, J. Mod. Dyn., 5 (2011), 747. doi: 10.3934/jmd.2011.5.747. [20] G. Hector, Feuilletages en cylindres,, in Geometry and Topology (Proc. III Latin Amer. School of Math., (1976), 252. [21] K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 21 (2001), 801. doi: 10.1017/S0143385701001390. [22] B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). [23] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds,, Bull. Amer. Math. Soc., 76 (1970), 1015. 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. 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. [27] K. C. Millett, Compact foliations,, in Differential Topology and Geometry (Proc. Colloq., (1974), 277. [28] S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761. 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. doi: 10.2307/1968772. [30] R. Potrie and A. Hammerlindl, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group,, , (2013). [31] C. Pugh and M. Shub, Ergodicity of Anosov actions,, Invent. Math., 15 (1972), 1. doi: 10.1007/BF01418639. [32] G. Reeb, Sur Certaines Propriétés Topologiques des Variétés Feuilletées,, Publ. Inst. Math. Univ. Strasbourg 11, (1183), 5. [33] Walter Rudin, Real and Complex Analysis,, Third edition, (1987). [34] D. Sullivan, A counterexample to the periodic orbit conjecture,, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 5. [35] D. Sullivan, A new flow,, Bull. Amer. Math. Sco., 82 (1976), 331. 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. [37] B. von Kerékjártó, Über Transformationen des ebenen Kreisringes,, Mathematische Annalen, 80 (1919), 33. [38] A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow,, Ergodic Theory Dynam. Systems, 18 (1998), 1545. doi: 10.1017/S0143385798117984.
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