October  2011, 5(4): 747-769. doi: 10.3934/jmd.2011.5.747

Partially hyperbolic diffeomorphisms with compact center foliations

1. 

Department ofMathematical Sciences, The State University of New York, Binghamton, NY 13902, United States

Received  November 2011 Revised  February 2012 Published  March 2012

Let $f\colon M\to M$ be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of $f$. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves.
    Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:
  1. 1. the center foliation of $f$ has codimension 2, or
  2. 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is one-dimensional.
Citation: Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747
References:
[1]

V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep., 75 (1981), 287-325. doi: 10.1016/0370-1573(81)90186-1.

[2]

D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Thesis, 2011.

[3]

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.

[4]

R. Bowen, Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.

[5]

M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, in "Modern Dynamical Systems and Applications," 307-312, Cambridge Univ. Press, Cambridge, 2004.

[6]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. doi: 10.3934/jmd.2008.2.541.

[7]

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

[8]

P. Carrasco, "Compact Dynamical Foliations," Thesis, 2010.

[9]

Y. Coudene, Pictures of hyperbolic dynamical systems, Notices Amer. Math. Soc., 53 (2006), 8-13.

[10]

R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 13-32. doi: 10.1016/0040-9383(77)90028-3.

[11]

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

[12]

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

[13]

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.

[14]

J. Franks, Anosov diffeomorphisms, in "1970 Global Analysis" (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 61-93.

[15]

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.

[16]

J. G. Hocking and G. S. Young, "Topology," Second edition, Dover Publications, Inc., New York, 1988.

[17]

R. Langevin, A list of questions about foliations, in "Differential Topology, Foliations, and Group Actions" (Rio de Janeiro, 1992), Contemp. Math., 161, Amer. Math. Soc., Providence, RI, (1994), 59-80.

[18]

D. Montgomery, Pointwise periodic homeomorphisms, Amer. J. Math., 59 (1937), 118-120. doi: 10.2307/2371565.

[19]

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

[20]

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

[21]

, F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,, private communication., (). 

[22]

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

[23]

E. Vogt, Foliations of codimension 2 with all leaves compact, Manuscripta Math., 18 (1976), 187-212. doi: 10.1007/BF01184305.

show all references

References:
[1]

V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep., 75 (1981), 287-325. doi: 10.1016/0370-1573(81)90186-1.

[2]

D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Thesis, 2011.

[3]

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.

[4]

R. Bowen, Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.

[5]

M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, in "Modern Dynamical Systems and Applications," 307-312, Cambridge Univ. Press, Cambridge, 2004.

[6]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. doi: 10.3934/jmd.2008.2.541.

[7]

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

[8]

P. Carrasco, "Compact Dynamical Foliations," Thesis, 2010.

[9]

Y. Coudene, Pictures of hyperbolic dynamical systems, Notices Amer. Math. Soc., 53 (2006), 8-13.

[10]

R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 13-32. doi: 10.1016/0040-9383(77)90028-3.

[11]

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

[12]

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

[13]

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.

[14]

J. Franks, Anosov diffeomorphisms, in "1970 Global Analysis" (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 61-93.

[15]

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.

[16]

J. G. Hocking and G. S. Young, "Topology," Second edition, Dover Publications, Inc., New York, 1988.

[17]

R. Langevin, A list of questions about foliations, in "Differential Topology, Foliations, and Group Actions" (Rio de Janeiro, 1992), Contemp. Math., 161, Amer. Math. Soc., Providence, RI, (1994), 59-80.

[18]

D. Montgomery, Pointwise periodic homeomorphisms, Amer. J. Math., 59 (1937), 118-120. doi: 10.2307/2371565.

[19]

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

[20]

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

[21]

, F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,, private communication., (). 

[22]

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

[23]

E. Vogt, Foliations of codimension 2 with all leaves compact, Manuscripta Math., 18 (1976), 187-212. doi: 10.1007/BF01184305.

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