# American Institue of Mathematical Sciences

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

## Hölder foliations, revisited

 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.
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), 90 (1967). [2] D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy,", Ph.D Thesis, (2011). [3] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds,, Topology, 44 (2005), 475. 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. doi: 10.3934/dcds.2008.22.89. [5] P. Carrasco, "Compact Dynamical Foliations,", Ph.D Thesis, (2010). [6] J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry,", North-Holland Mathematical Library, (1975). [7] D. Chillingworth, unpublished,, circa 1970., (1970).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. 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. [10] D. Epstein, Foliations with all leaves compact,, Ann. Inst. Fourier (Grenoble), 26 (1976), 265. doi: 10.5802/aif.607. [11] A. Hammerlindl, Quasi-isometry and plaque expansiveness,, Canadian Mathematical Bulletin, 54 (2011), 676. doi: 10.4153/CMB-2011-024-7. [12] B. Hasselblatt, Regularity of the Anosov splitting. II,, Ergodic Theory Dynam. Systems, 17 (1997), 169. doi: 10.1017/S0143385797069757. [13] B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations,, Ergodic Theory Dynam. Systems, 19 (1999), 643. doi: 10.1017/S0143385799133868. [14] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics,, in, 51 (2007), 35. [15] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (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, 54 (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. [19] C. Pugh, M. Shub and A. Wilkinson, Hölder foliations,, Duke Math. J., 86 (1997), 517. 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. [21] J. Schmeling and Ra. Siegmund-Schultze, Hölder-continuity of the holonomy maps for hyperbolic sets,, in, 1514 (1992), 174. [22] M. Shub, "Global Stability of Dynamical Systems,", With the collaboration of Albert Fathi and Rémi Langevin, (1987). [23] A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow,, Ergod. Th. & Dynam. Sys., 18 (1998), 1545. [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), 90 (1967). [2] D. Bohnet, "Partially Hyperbolic Systems with a Compact Center Foliation with Finite Holonomy,", Ph.D Thesis, (2011). [3] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds,, Topology, 44 (2005), 475. 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. doi: 10.3934/dcds.2008.22.89. [5] P. Carrasco, "Compact Dynamical Foliations,", Ph.D Thesis, (2010). [6] J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry,", North-Holland Mathematical Library, (1975). [7] D. Chillingworth, unpublished,, circa 1970., (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. 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. [10] D. Epstein, Foliations with all leaves compact,, Ann. Inst. Fourier (Grenoble), 26 (1976), 265. doi: 10.5802/aif.607. [11] A. Hammerlindl, Quasi-isometry and plaque expansiveness,, Canadian Mathematical Bulletin, 54 (2011), 676. doi: 10.4153/CMB-2011-024-7. [12] B. Hasselblatt, Regularity of the Anosov splitting. II,, Ergodic Theory Dynam. Systems, 17 (1997), 169. doi: 10.1017/S0143385797069757. [13] B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations,, Ergodic Theory Dynam. Systems, 19 (1999), 643. doi: 10.1017/S0143385799133868. [14] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics,, in, 51 (2007), 35. [15] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (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, 54 (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. [19] C. Pugh, M. Shub and A. Wilkinson, Hölder foliations,, Duke Math. J., 86 (1997), 517. 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. [21] J. Schmeling and Ra. Siegmund-Schultze, Hölder-continuity of the holonomy maps for hyperbolic sets,, in, 1514 (1992), 174. [22] M. Shub, "Global Stability of Dynamical Systems,", With the collaboration of Albert Fathi and Rémi Langevin, (1987). [23] A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow,, Ergod. Th. & Dynam. Sys., 18 (1998), 1545. [24] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, preprint, (2008).
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