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Partially hyperbolic diffeomorphisms with compact center foliations

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  • 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.
    Mathematics Subject Classification: Primary: 37D30, 57R30; Secondary: 37D20.

    Citation:

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  • [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]
    [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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