Hölder foliations, revisited

Previous Article
Genericity of nonuniform hyperbolicity in dimension 3
JMD Home
This Issue
Next Article
Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

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

Abstract
Related Papers

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.
[2]	D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Ph.D Thesis, Universität Hamburg, 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]	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.
[5]	P. Carrasco, "Compact Dynamical Foliations," Ph.D Thesis, University of Toronto, Canada, 2010.
[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.
[7]	D. Chillingworth, unpublished, circa 1970.
[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.
[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, 4405-4430.
[10]	D. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265-282. doi: 10.5802/aif.607.
[11]	A. Hammerlindl, Quasi-isometry and plaque expansiveness, Canadian Mathematical Bulletin, 54 (2011), 676-679. doi: 10.4153/CMB-2011-024-7.
[12]	B. Hasselblatt, Regularity of the Anosov splitting. II, Ergodic Theory Dynam. Systems, 17 (1997), 169-172. doi: 10.1017/S0143385797069757.
[13]	B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Ergodic Theory Dynam. Systems, 19 (1999), 643-656. doi: 10.1017/S0143385799133868.
[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.
[15]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[16]	Y. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained, preprint.
[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.
[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.
[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.
[20]	C. Pugh, M. Shub and A. Wilkinson, Correction to: "Hölder foliations," Duke Math. J., 105 (2000), 105-106.
[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.
[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.
[23]	A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergod. Th. & Dynam. Sys., 18 (1998), 1545-1587.
[24]	A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, preprint, 2008.

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.
[2]	D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy," Ph.D Thesis, Universität Hamburg, 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]	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.
[5]	P. Carrasco, "Compact Dynamical Foliations," Ph.D Thesis, University of Toronto, Canada, 2010.
[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.
[7]	D. Chillingworth, unpublished, circa 1970.
[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.
[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, 4405-4430.
[10]	D. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26 (1976), 265-282. doi: 10.5802/aif.607.
[11]	A. Hammerlindl, Quasi-isometry and plaque expansiveness, Canadian Mathematical Bulletin, 54 (2011), 676-679. doi: 10.4153/CMB-2011-024-7.
[12]	B. Hasselblatt, Regularity of the Anosov splitting. II, Ergodic Theory Dynam. Systems, 17 (1997), 169-172. doi: 10.1017/S0143385797069757.
[13]	B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Ergodic Theory Dynam. Systems, 19 (1999), 643-656. doi: 10.1017/S0143385799133868.
[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.
[15]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[16]	Y. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained, preprint.
[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.
[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.
[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.
[20]	C. Pugh, M. Shub and A. Wilkinson, Correction to: "Hölder foliations," Duke Math. J., 105 (2000), 105-106.
[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.
[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.
[23]	A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergod. Th. & Dynam. Sys., 18 (1998), 1545-1587.
[24]	A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, preprint, 2008.

[1]	Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81
[2]	Boris Hasselblatt. Critical regularity of invariant foliations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931
[3]	Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[4]	Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[5]	Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[6]	Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[7]	Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012
[8]	Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[9]	Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
[10]	Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227
[11]	Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641
[12]	Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[13]	Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029
[14]	Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55
[15]	Jun Shen, Kening Lu, Bixiang Wang. Invariant manifolds and foliations for random differential equations driven by colored noise. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6201-6246. doi: 10.3934/dcds.2020276
[16]	Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175
[17]	Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819
[18]	Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123
[19]	Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
[20]	Radu Saghin. Note on homology of expanding foliations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 349-360. doi: 10.3934/dcdss.2009.2.349

2021 Impact Factor: 0.641

Tools

Metrics

Other articles
by authors

on AIMS
Charles Pugh Michael Shub Amie Wilkinson
on Google Scholar
Charles Pugh Michael Shub Amie Wilkinson

[Back to Top]