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January  2012, 6(1): 79-120. doi: 10.3934/jmd.2012.6.79

Hölder foliations, revisited

Charles Pugh 1, , Michael Shub 2, and  Amie Wilkinson 1,

1. 

Department ofMathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, United States, United States
2. 

CONICET, IMAS, Universidad de Buenos Aires, Buenos Aires, Argentina

Received  December 2011 Published  May 2012

We investigate transverse Hölder regularity of some canonical leaf conjugacies in normally hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.
Keywords: normal hyperbolicity, invariant foliations, H\"older regularity of foliations., Partial hyperbolicity.
Mathematics Subject Classification: 37D3.
Citation: Charles Pugh, Michael Shub, Amie Wilkinson. Hölder foliations, revisited. Journal of Modern Dynamics, 2012, 6 (1) : 79-120. doi: 10.3934/jmd.2012.6.79
References:
[1]

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, (Russian), Trudy Mat. Inst. Stecklov., 90 (1967), 209 pp.  Google Scholar

[2]

D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Ph.D Thesis, Universität Hamburg, 2011. Google Scholar

[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.  Google Scholar

[4]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst., 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[5]

P. Carrasco, "Compact Dynamical Foliations," Ph.D Thesis, University of Toronto, Canada, 2010. Google Scholar

[6]

J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry," North-Holland Mathematical Library, Vol. 9, North Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing, Co., Inc., New York, 1975.  Google Scholar

[7]

D. Chillingworth, unpublished, circa 1970. Google Scholar

[8]

D. Damjanović and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions, Discrete Contin. Dyn. Syst., 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985.  Google Scholar

[9]

D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl chamber flows on $SL(n,\mathbbR)$/$Gamma$,, Int. Math. Res. Not. IMRN, 2011 (): 4405.   Google Scholar

[10]

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

[11]

A. Hammerlindl, Quasi-isometry and plaque expansiveness, Canadian Mathematical Bulletin, 54 (2011), 676-679. doi: 10.4153/CMB-2011-024-7.  Google Scholar

[12]

B. Hasselblatt, Regularity of the Anosov splitting. II, Ergodic Theory Dynam. Systems, 17 (1997), 169-172. doi: 10.1017/S0143385797069757.  Google Scholar

[13]

B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Ergodic Theory Dynam. Systems, 19 (1999), 643-656. doi: 10.1017/S0143385799133868.  Google Scholar

[14]

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

[15]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[16]

Y. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained,, preprint., ().   Google Scholar

[17]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

V. Niţică and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995), 751-810.  Google Scholar

[19]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546. doi: 10.1215/S0012-7094-97-08616-6.  Google Scholar

[20]

C. Pugh, M. Shub and A. Wilkinson, Correction to: "Hölder foliations," Duke Math. J., 105 (2000), 105-106.  Google Scholar

[21]

J. Schmeling and Ra. Siegmund-Schultze, Hölder-continuity of the holonomy maps for hyperbolic sets, in "Ergodic Theory and Related Topics, III" (Güstrow, 1990), Lecture Notes in Math., 1514, Springer, Berlin, (1992), 174-191.  Google Scholar

[22]

M. Shub, "Global Stability of Dynamical Systems," With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy, Springer-Verlag, New York, 1987.  Google Scholar

[23]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergod. Th. & Dynam. Sys., 18 (1998), 1545-1587.  Google Scholar

[24]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, preprint, 2008. Google Scholar

show all references

References:
[1]

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, (Russian), Trudy Mat. Inst. Stecklov., 90 (1967), 209 pp.  Google Scholar

[2]

D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Ph.D Thesis, Universität Hamburg, 2011. Google Scholar

[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.  Google Scholar

[4]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst., 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[5]

P. Carrasco, "Compact Dynamical Foliations," Ph.D Thesis, University of Toronto, Canada, 2010. Google Scholar

[6]

J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry," North-Holland Mathematical Library, Vol. 9, North Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing, Co., Inc., New York, 1975.  Google Scholar

[7]

D. Chillingworth, unpublished, circa 1970. Google Scholar

[8]

D. Damjanović and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions, Discrete Contin. Dyn. Syst., 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985.  Google Scholar

[9]

D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl chamber flows on $SL(n,\mathbbR)$/$Gamma$,, Int. Math. Res. Not. IMRN, 2011 (): 4405.   Google Scholar

[10]

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

[11]

A. Hammerlindl, Quasi-isometry and plaque expansiveness, Canadian Mathematical Bulletin, 54 (2011), 676-679. doi: 10.4153/CMB-2011-024-7.  Google Scholar

[12]

B. Hasselblatt, Regularity of the Anosov splitting. II, Ergodic Theory Dynam. Systems, 17 (1997), 169-172. doi: 10.1017/S0143385797069757.  Google Scholar

[13]

B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Ergodic Theory Dynam. Systems, 19 (1999), 643-656. doi: 10.1017/S0143385799133868.  Google Scholar

[14]

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

[15]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[16]

Y. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained,, preprint., ().   Google Scholar

[17]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

V. Niţică and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995), 751-810.  Google Scholar

[19]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546. doi: 10.1215/S0012-7094-97-08616-6.  Google Scholar

[20]

C. Pugh, M. Shub and A. Wilkinson, Correction to: "Hölder foliations," Duke Math. J., 105 (2000), 105-106.  Google Scholar

[21]

J. Schmeling and Ra. Siegmund-Schultze, Hölder-continuity of the holonomy maps for hyperbolic sets, in "Ergodic Theory and Related Topics, III" (Güstrow, 1990), Lecture Notes in Math., 1514, Springer, Berlin, (1992), 174-191.  Google Scholar

[22]

M. Shub, "Global Stability of Dynamical Systems," With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy, Springer-Verlag, New York, 1987.  Google Scholar

[23]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergod. Th. & Dynam. Sys., 18 (1998), 1545-1587.  Google Scholar

[24]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, preprint, 2008. Google Scholar

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