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Ziggurats and rotation numbers
Partially hyperbolic diffeomorphisms with compact center foliations
1.  Department ofMathematical Sciences, The State University of New York, Binghamton, NY 13902, United States 
Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:
 1. the center foliation of $f$ has codimension 2, or
 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is onedimensional.
References:
[1] 
V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep., 75 (1981), 287325. doi: 10.1016/03701573(81)901861. 
[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 3manifolds, Topology, 44 (2005), 475508. 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), 377397. 
[5] 
M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3manifolds with commutative fundamental group, in "Modern Dynamical Systems and Applications," 307312, Cambridge Univ. Press, Cambridge, 2004. 
[6] 
D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541580. 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]  
[9] 
Y. Coudene, Pictures of hyperbolic dynamical systems, Notices Amer. Math. Soc., 53 (2006), 813. 
[10] 
R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 1332. doi: 10.1016/00409383(77)900283. 
[11] 
D. B. A. Epstein, Periodic flows on threemanifolds, Ann. of Math. (2), 95 (1972), 6682. doi: 10.2307/1970854. 
[12] 
D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265282. 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), 539552. 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., 6193. 
[15] 
K. Hiraide, A simple proof of the FranksNewhouse theorem on codimensionone Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 801806. 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), 5980. 
[18] 
D. Montgomery, Pointwise periodic homeomorphisms, Amer. J. Math., 59 (1937), 118120. doi: 10.2307/2371565. 
[19] 
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761770. 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), 3587. 
[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), 514. 
[23] 
E. Vogt, Foliations of codimension 2 with all leaves compact, Manuscripta Math., 18 (1976), 187212. 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), 287325. doi: 10.1016/03701573(81)901861. 
[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 3manifolds, Topology, 44 (2005), 475508. 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), 377397. 
[5] 
M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3manifolds with commutative fundamental group, in "Modern Dynamical Systems and Applications," 307312, Cambridge Univ. Press, Cambridge, 2004. 
[6] 
D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541580. 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]  
[9] 
Y. Coudene, Pictures of hyperbolic dynamical systems, Notices Amer. Math. Soc., 53 (2006), 813. 
[10] 
R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 1332. doi: 10.1016/00409383(77)900283. 
[11] 
D. B. A. Epstein, Periodic flows on threemanifolds, Ann. of Math. (2), 95 (1972), 6682. doi: 10.2307/1970854. 
[12] 
D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265282. 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), 539552. 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., 6193. 
[15] 
K. Hiraide, A simple proof of the FranksNewhouse theorem on codimensionone Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 801806. 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), 5980. 
[18] 
D. Montgomery, Pointwise periodic homeomorphisms, Amer. J. Math., 59 (1937), 118120. doi: 10.2307/2371565. 
[19] 
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761770. 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), 3587. 
[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), 514. 
[23] 
E. Vogt, Foliations of codimension 2 with all leaves compact, Manuscripta Math., 18 (1976), 187212. doi: 10.1007/BF01184305. 
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