Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Regular points for ergodic Sinaĭ measures

Author: Masato Tsujii
Journal: Trans. Amer. Math. Soc. 328 (1991), 747-766
MSC: Primary 58F11
MathSciNet review: 1072103
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure $ \mu $ if it is generic for $ \mu $ and the Lyapunov exponents at it coincide with those of $ \mu $. We show that an ergodic measure with no zero Lyapunov exponent is absolutely continuous with respect to unstable foliation $ [$L$ ]$ if and only if the set of all points which are regular for it has positive Lebesgue measure.

References [Enhancements On Off] (What's this?)

  • [B] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphism, Lecture Notes in Math., vol. 470, Springer, New York, 1975. MR 0442989 (56:1364)
  • [GH] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector field, Appl. Math. Sci., vol. 42, Springer, New York, 1986. MR 709768 (85f:58002)
  • [K] A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphism, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137-173. MR 573822 (81i:28022)
  • [KS] A. B. Katok and J. P. Strelcyn, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Math., vol. 1222, Springer, New York, 1986. MR 872698 (88k:58075)
  • [HPS] M. W. Hirsh, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer, New York, 1977. MR 0501173 (58:18595)
  • [L] F. Ledrappier, Propriétés ergodique des mesures de Sinaĭ, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 163-188. MR 743818 (86f:58092)
  • [M] R. Mañè, Ergodic theory and differentiable dynamics, Springer-Verlag, New York, 1987. MR 889254 (88c:58040)
  • [O] V. I. Oseledec, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-231. MR 0240280 (39:1629)
  • [P1] Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv. 10 (1976), no. 6, 1261-1305.
  • [P2] -, Lyapunov characteristic exponent and smooth ergodic theory, Russian Math. Surveys 32 (1977), no. 4, 55-114. MR 0466791 (57:6667)
  • [PS] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989). MR 983869 (90h:58057)
  • [R1] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27-58. MR 556581 (81f:58031)
  • [R2] -, Sensitive dependence on initial conditions and turbulent behavior of dynamical systems, Ann. New York Acad. Sci. 316 (1979), 408-416. MR 556846 (82b:58048)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F11

Retrieve articles in all journals with MSC: 58F11

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society