Prevalence of nonLipschitz Anosov foliations
Authors:
Boris Hasselblatt and Amie Wilkinson
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 9398
MSC (1991):
Primary 58F15; Secondary 53C12
Published electronically:
September 11, 1997
MathSciNet review:
1465582
Fulltext PDF Free Access
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Abstract: We give sharp regularity results for the invariant subbundles of hyperbolic dynamical systems and give open dense sets of codimension one systems where this regularity is not exceeded as well as open dense sets of symplectic, geodesic, and codimension one systems where the analogous regularity results of Pugh, Shub, and Wilkinson are optimal. We produce open sets of symplectic Anosov diffeomorphisms and flows with low transverse Hölder regularity of the invariant foliations almost everywhere. Prevalence of low regularity of conjugacies on large sets is a corollary. We also establish a new connection between the transverse regularity of foliations and their tangent subbundles.
 [A1]
D.
V. Anosov, Geodesic flows on closed Riemannian manifolds of
negative curvature, Trudy Mat. Inst. Steklov. 90
(1967), 209 (Russian). MR 0224110
(36 #7157)
D.
V. Anosov, Geodesic flows on closed Riemann manifolds with negative
curvature., Proceedings of the Steklov Institute of Mathematics, No.
90 (1967). Translated from the Russian by S. Feder, American Mathematical
Society, Providence, R.I., 1969. MR 0242194
(39 #3527)
 [A2]
D.
V. Anosov, Tangential fields of transversal foliations in
𝑈systems, Mat. Zametki 2 (1967),
539–548 (Russian). MR 0242190
(39 #3523)
 [BCG]
Gérard
Besson, Gilles
Courtois, and Sylvestre
Gallot, Minimal entropy and Mostow’s rigidity theorems,
Ergodic Theory Dynam. Systems 16 (1996), no. 4,
623–649. MR 1406425
(97e:58177), http://dx.doi.org/10.1017/S0143385700009019
 [F]
Neil
Fenichel, Asymptotic stability with rate conditions, Indiana
Univ. Math. J. 23 (1973/74), 1109–1137. MR 0339276
(49 #4036)
Neil
Fenichel, Asymptotic stability with rate conditions. II,
Indiana Univ. Math. J. 26 (1977), no. 1, 81–93.
MR
0426056 (54 #14002)
 [GPS]
Matthew
Grayson, Charles
Pugh, and Michael
Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2)
140 (1994), no. 2, 295–329. MR 1298715
(95g:58128), http://dx.doi.org/10.2307/2118602
 [G]
Leon
W. Green, The generalized geodesic flow, Duke Math. J.
41 (1974), 115–126; correction, ibid. 42\ (1975),
381. MR
0370659 (51 #6885)
 [H1]
Boris
Hasselblatt, Regularity of the Anosov splitting and of horospheric
foliations, Ergodic Theory Dynam. Systems 14 (1994),
no. 4, 645–666. MR 1304137
(95j:58130), http://dx.doi.org/10.1017/S0143385700008105
 [H2]
Boris
Hasselblatt, Horospheric foliations and relative pinching, J.
Differential Geom. 39 (1994), no. 1, 57–63. MR 1258914
(95c:58137)
 [H3]
Boris
Hasselblatt, Periodic bunching and invariant foliations, Math.
Res. Lett. 1 (1994), no. 5, 597–600. MR 1295553
(95h:58097), http://dx.doi.org/10.4310/MRL.1994.v1.n5.a7
 [H4]
Boris Hasselblatt, Regularity of the Anosov splitting II, Ergodic Theory and Dynamical Systems, 17 (1997), 169172. CMP 97:10
 [HPS]
M.
W. Hirsch, C.
C. Pugh, and M.
Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol.
583, SpringerVerlag, BerlinNew York, 1977. MR 0501173
(58 #18595)
 [HK]
S.
Hurder and A.
Katok, Differentiability, rigidity and GodbillonVey classes for
Anosov flows, Inst. Hautes Études Sci. Publ. Math.
72 (1990), 5–61 (1991). MR 1087392
(92b:58179)
 [KH]
Anatole
Katok and Boris
Hasselblatt, Introduction to the modern theory of dynamical
systems, Encyclopedia of Mathematics and its Applications,
vol. 54, Cambridge University Press, Cambridge, 1995. With a
supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
(96c:58055)
 [LM]
R.
de la Llave and R.
Moriyón, Invariants for smooth conjugacy of hyperbolic
dynamical systems. IV, Comm. Math. Phys. 116 (1988),
no. 2, 185–192. MR 939045
(90h:58064)
 [N]
S.
E. Newhouse, On codimension one Anosov diffeomorphisms, Amer.
J. Math. 92 (1970), 761–770. MR 0277004
(43 #2741)
 [P]
Ja.
B. Pesin, The existence of invariant foliations for a
diffeomorphism of a smooth manifold, Mat. Sb. (N.S.)
91(133) (1973), 202–210, 287 (Russian). MR 0343307
(49 #8049)
 [PS]
Charles Pugh, Michael Shub, Stably ergodic dynamical systems and partial hyperbolicity, Journal of Complexity, to appear
 [PSW]
Charles Pugh, Michael Shub, Amie Wilkinson, Hölder foliations, Duke Mathematical Journal, 86 (1997), no. 3, 517546. CMP 97:07
 [SS]
J.
Schmeling and Ra.
SiegmundSchultze, Hölder continuity of the holonomy maps for
hyperbolic basic sets. I, Ergodic theory and related topics, III
(Güstrow, 1990) Lecture Notes in Math., vol. 1514, Springer,
Berlin, 1992, pp. 174–191. MR 1179182
(93j:58104), http://dx.doi.org/10.1007/BFb0097538
 [W]
Amie Wilkinson, Stable ergodicity of the timeone map of a geodesic flow, Ergodic Theory and Dynamical Systems, to appear
 [A1]
 D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. 90 (1967). MR 36:7157, MR 39:3527
 [A2]
 D. V. Anosov, Tangential fields of transversal foliations in ``Usystems'', Mat. Zametki 2 (1967), no. 5, 818823. MR 39:3523
 [BCG]
 Gérard Besson, Gilles Courtois, Sylvestre Gallot, Minimal entropy and Mostow's rigidity theorems, Ergodic Theory and Dynamical Systems 16 (1996), no. 4, 623649. MR 97e:58177
 [F]
 Neil Fenichel, Asymptotic stability with rate conditions, Indiana University Math. Journal 23 (1974), 11091137; 26 (1977), no. 1, 8193. MR 49:4036, MR 54:14002
 [GPS]
 Matthew Grayson, Charles Pugh, Michael Shub, Stably ergodic diffeomorphisms, Annals of Mathematics (2) 140 (1994), no. 2, 295329. MR 95g:58128
 [G]
 Leon W. Green, The generalized geodesic flow, Duke Mathematical Journal 41 (1974), 115126. MR 51:6885
 [H1]
 Boris Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory and Dynamical Systems, 14 (1994), no. 4, 645666. MR 95j:58130
 [H2]
 Boris Hasselblatt, Horospheric foliations and relative pinching, Journal of Differential Geometry 39 (1994), no. 1, 5763. MR 95c:58137
 [H3]
 Boris Hasselblatt, Periodic bunching and invariant foliations, Mathematical Research Letters 1 (1994), no. 5, 597600. MR 95h:58097
 [H4]
 Boris Hasselblatt, Regularity of the Anosov splitting II, Ergodic Theory and Dynamical Systems, 17 (1997), 169172. CMP 97:10
 [HPS]
 Morris Hirsch, Charles Pugh, Michael Shub, Invariant manifolds, Lecture Notes in Mathematics 583, SpringerVerlag, 1977. MR 58:18595
 [HK]
 Steven Hurder, Anatole Katok, Differentiability, rigidity, and GodbillonVey classes for Anosov flows, Publications IHES 72 (1990), 561. MR 92b:58179
 [KH]
 Anatole Katok, Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995. MR 96c:58055
 [LM]
 Rafael de la Llave, Roberto Moriyon, Invariants for smooth conjugacy of hyperbolic dynamical systems. IV, Communications in Mathematical Physics 116 (1988), no. 2, 185192. MR 90h:58064
 [N]
 Sheldon Newhouse, On codimension one Anosov diffeomorphisms, American Journal of Mathematics 92 (1970), 761770. MR 43:2741
 [P]
 Yakov Pesin, On the existence of invariant fiberings for a diffeomorphism of a smooth manifold, Math. USSR Sbornik 20 (1973), no. 2, 213222. MR 49:8049
 [PS]
 Charles Pugh, Michael Shub, Stably ergodic dynamical systems and partial hyperbolicity, Journal of Complexity, to appear
 [PSW]
 Charles Pugh, Michael Shub, Amie Wilkinson, Hölder foliations, Duke Mathematical Journal, 86 (1997), no. 3, 517546. CMP 97:07
 [SS]
 Jörg Schmeling, Rainer SiegmundSchulze, Hölder continuity of the holonomy maps for hyperbolic basic sets, I, Ergodic theory and related topics, III (Güstrow, 1990), pp. 174191, Springer lecture notes in mathematics 1514, Springer, Berlin, 1992. MR 93j:58104
 [W]
 Amie Wilkinson, Stable ergodicity of the timeone map of a geodesic flow, Ergodic Theory and Dynamical Systems, to appear
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Additional Information
Boris Hasselblatt
Affiliation:
Department of Mathematics\ Tufts University\ Medford, MA 021555597
Email:
bhasselb@tufts.edu
Amie Wilkinson
Affiliation:
Department of Mathematics\ Northwestern University\ Evanston, IL 602082730
Email:
wilkinso@math.nwu.edu
DOI:
http://dx.doi.org/10.1090/S1079676297000309
PII:
S 10796762(97)000309
Keywords:
Anosov system,
hyperbolic system,
invariant foliations,
stable foliation,
Anosov splitting,
horospheric foliations,
holonomy,
H\"older structures,
conjugacy
Received by editor(s):
May 9, 1997
Published electronically:
September 11, 1997
Dedicated:
To the memory of Gunnar Hasselblatt, 19.8.1928–12.7.1997
Communicated by:
Krystyna Kuperberg
Article copyright:
© Copyright 1997
American Mathematical Society
