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Smoothness of stable holonomies inside center-stable manifolds

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  18 October 2021

AARON BROWN
AARON BROWN*
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL60208, USA
*
(e-mail: awb@northwestern.edu)
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Abstract

Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for $C^{1+\beta }$ diffeomorphisms are uniformly bi-Lipschitz and, in fact, $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ . This verifies the ergodicity of suitably center-bunched, essentially accessible, partially hyperbolic $C^{1+\beta }$ diffeomorphisms and verifies that the Ledrappier–Young entropy formula holds for $C^{1+\beta }$ diffeomorphisms of compact manifolds.

Keywords

holonomiespartial hyperbolicityergodicityentropy

MSC classification

Secondary: 37C40: Smooth ergodic theory, invariant measures 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3593 - 3618
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Barreira, L., Pesin, Y. and Schmeling, J.. Dimension and product structure of hyperbolic measures. Ann. of Math. (2) 149 (1999), 755–783.CrossRefGoogle Scholar
Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. (2) 171 (2010), 451–489.CrossRefGoogle Scholar
Burns, K. and Wilkinson, A.. A note on stable holonomy between centers. Preprint.Google Scholar
Fathi, A., Herman, M.-R. and Yoccoz, J.-C.. A proof of Pesin’s stable manifold theorem. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 177–215.Google Scholar
Grayson, M., Pugh, C. and Shub, M.. Stably ergodic diffeomorphisms. Ann. of Math. (2) 140 (1994), 295–329.CrossRefGoogle Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
Journé, J.-L.. A regularity lemma for functions of several variables. Rev. Mat. Iberoam. 4 (1988), 187–193.CrossRefGoogle Scholar
Ledrappier, F.. Propriétés ergodiques des mesures de Sinaï. Publ. Math. Inst. Hautes Études Sci. 59 (1984), 163–188 (in French).CrossRefGoogle Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122 (1985), 509–539.CrossRefGoogle Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122 (1985), 540–574.CrossRefGoogle Scholar
Pugh, C. and Shub, M.. Ergodic attractors. Trans. Amer. Math. Soc. 312 (1989), 1–54.CrossRefGoogle Scholar
Pugh, C. and Shub, M.. Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. (JEMS) 2 (2000), 1–52.CrossRefGoogle Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86 (1997), 517–546.CrossRefGoogle Scholar