Abstract
In the space of diffeomorphisms of an arbitrary closed manifold of dimension ≥ 3, we construct an open set such that each difteomorphism in this set has an invariant ergodic measure with respect to which one of its Lyapunov exponents is zero. These difteomorphisins are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with the circle as the fiber. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Abraham and S. Smale, “Nongenericity of Ω-stability,” in: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 5–8.
Ya. B. Pesiii, “Characteristic Lyapunov exponents and smooth ergodic theory,” Uspekhi Mat. Nauk, 32:4(196) (1977), 55–112; English transl.: Russian Math. Surveys, 32:4 (1977), 55–114.
H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Statist., 31 (1960), 457–469.
G. Bonatti, X. Gomez-Mont, and M. Viana, “Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices,” Ann. Inst. H. Poincaré, Anal. Non Linéaire, 20:4 (2003), 579–624.
L. Arnold, N. D. Gong, and V. I. Oseledets, “Jordan normal form for linear cocycles,” Random Oper. Stochastic Equations, 7:4 (1999), 303–358.
L. Arnold and N. D. Gong, “Linear cocycles with simple spectrum are dense in L ∞, ” Ergodic Theory Dynam. Systems, 19:6 (1999), 1389–1404.
J. Bochi, “Genericity of zero Lyapunov exponents,” Ergodic Theory Dynam. Systems, 22:6 (2002), 1667–1696.
J. Bochi and M. Viana, “Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps,” Ann. Inst. H. Poincaré, Anal. Non Linéaire, 19:1 (2002), 113–123.
A. Baraviera and G. Bonatti, “Removing zero Lyapunov exponents,” Ergodic Theory Dynam. Systems, 23 (2003), 1655–1670.
M. Shub and A. Wilkinson, “Pathological foliations and removable zero exponents,” Invent. Math., 139 (2000), 495–508.
D. Ruelle and A. Wilkinson, “Absolutely Singular dynamical foliations,” Comm. Math. Phys., 219:3 (2001), 481–487.
D. Dolgopyat and Y. Pesin, “Every compact manifold carries a completely hyperbolic diffeomorphism,” Ergodic Theory Dynam. Systems, 22:2 (2002), 409–435.
M. W. Hirsch, G. G. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin-New York, 1977.
A. S. Gorodetski and Yu. S. Ilyashenko, “Certain new robust properties of invariant sets and attractors of dynamical Systems,” Funkts. Anal. Prilozhen., 33:2 (1999), 16–30; English transl.: Functional Anal. Appl., 33:2 (1999), 95–105.
A. S. Gorodetski and Yu. S. Ilyashenko, “Some properties of skew products over the horseshoe and solenoid,” Trudy Mat. Inst. Steklova, 231 (2000), 90–112; English transl.: Proc. Steklov Inst. Math., 231 (2000), 96–118.
A. S. Gorodetski, Minimal Attractors and Partially Hyperbolic Invariant Sets of Dynamical Systems, PhD thesis, Moscow State University, 2001.
A. S. Gorodetski, “The regularity of central leaves of partially hyperbolic sets and its applications,” Izv. Ross. Akad. Nauk Ser. Mat., 70:6 (2006), 19–44; English transl.: Russian Acad. Sci. Izv. Math., 70:6 (2006), 1093–1116.
A. S. Gorodetski, Yu. S. Ilyashenko, V. A. Kleptsyn, and M. B. Naisky, “Nonremovable zero Lyapunov exponents,” Funkts. Anal. Prilozhen., 39:1 (2005), 27–38; English transl.: Functional Anal. Appl., 39:1 (2005), 21–30.
Yu. Ilyashenko and W. Li, Nonlocal Bifurcations, Amer. Math. Soc., Providence, R.I., 1998.
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995.
A. B. Katok and A. M. Stepin, “Approximation of ergodic dynamical Systems by periodic transformations,” Dokl. Akad. Nauk SSSR, 171 (1966), 1268–1271; English transl.: Soviel Math. Dokl., 7 (1966), 1638–1641.
A. B. Katok and A. M. Stepin, “Approximations in ergodic theory,” Uspekhi Mat. Nauk, 22(137):5 (1967), 81–106; English transl.: Russian Math. Surveys, 22:5 (1967), 77–102.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 4, pp. 30–45, 2007
Original Russian Text Copyright © by V. A. Kleptsyn and M. B. Nalsky
Supported in part by CRDF grant RM1-2358-MO-02 and RFBR grants 07-01-00017-a and CNRS_a 05-01-02801. The first author was also supported by the Swiss National Science Foundation.
Rights and permissions
About this article
Cite this article
Kleptsyn, V.A., Nalsky, M.B. Persistence of nonhyperbolic measures for C 1-diffeomorphisms. Funct Anal Its Appl 41, 271–283 (2007). https://doi.org/10.1007/s10688-007-0025-8
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10688-007-0025-8