Abstract
A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Abraham and J. Robbins, Transversal Mappings and Flows (Benjamin, New York, 1967).
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov. Inst. Math. 90:000–000(1967).
M. Brin and Ja. Pesin, Partially hyperbolic dynamical systems, Math. USSR Izvestija 8:177–218 (1974).
D. Dolgopyat, On differentiability of SRB states, preprint.
M. Grayson, C. Pugh, and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2) 140:295–329 (1994).
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds. Lecture Notes in Mathematics, Vol.583 (Springer-Verlag, Berlin/New York, 1977).
A. Katok, G. Knieper, M. Pollicott, and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math. 98:581–597 (1989).
R. de la Llave, J. M. Marco, and R. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2) 123:537–611 (1986).
R. Mañê, On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps. Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol.898 (Springer, Berlin/New York, 1981).
J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, Math. Intelligencer 19:30–32 (1997).
C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity 13:125–179 (1997).
C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2:1–52 (2000).
D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187:227–241 (1997).
D. Ruelle, Extensions of a result by Shub and Wilkinson..., preprint
M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139:495–508 (2000).
A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergodic Theory Dynam. Systems 18:1545–1587 (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pugh, C., Shub, M. & Wilkinson, A. Partial Differentiability of Invariant Splittings. Journal of Statistical Physics 114, 891–921 (2004). https://doi.org/10.1023/B:JOSS.0000012511.80422.33
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000012511.80422.33