Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Next Article
     

Dimension of hyperbolic measures of random diffeomorphisms

Author(s): Pei-Dong Liu; Jian-Sheng Xie
Journal: Trans. Amer. Math. Soc. 358 (2006), 3751-3780.
MSC (2000): Primary 37C45; Secondary 37H15
Posted: April 11, 2006
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We consider dynamics of compositions of stationary random $ C^2$ diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.

References:

1.
L. Arnold. Random Dynamical Systems. Springer, 1998. MR 1723992 (2000m:37087)

2.
J. Bahnmüller and P.-D. Liu. Characterization of measures satisfying Pesin's entropy formula for random dynamical systems. J. Dynam. Diff. Equ. 10(3)(1998), 425-448. MR 1646606 (99j:58118)

3.
L. Barreira, Y. Pesin and J. Schmeling. Dimension and product structure of hyperbolic measures. Ann. Math. 149 (1999), 755-783. MR 1709302 (2000f:37027)

4.
T. Bogenschütz. Entropy, pressure, and a variational principle for random dynamical systems. Random and Computational Dynamics 1 (1992), 219-227. MR 1181382 (93k:28023)

5.
T. Bogenschütz. Equilibrium states for random dynamical systems. Ph.D. thesis, Institut für Dynamische Systeme, Universität Bremen (1993).

6.
M. de Guzmán. Differentiation of Integrals in $ \mathbf{R}^n$. Lecture Notes in Math. 481, Springer-Verlag, New York, 1975. MR 0457661 (56:15866)

7.
Y. Kifer. Ergodic Theory of Random Transformations. Birkhäuser, Boston, 1986. MR 0884892 (89c:58069)

8.
F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension. Ann. Math. 122 (1985), 540-574. MR 0819557 (87i:58101b)

9.
P.-D. Liu. (Survey) Dynamics of random transformations: smooth ergodic theory. Ergod. Th. Dynam. Sys. 21 (2001), 1279-1319. MR 1855833 (2002g:37024)

10.
P.-D. Liu. Random perturbations of Axiom A basic sets. J. Stat. Phys. 90(1/2) (1998), 467-490. MR 1611096 (99a:58108)

11.
P.-D. Liu and M. Qian. Smooth Ergodic Theory of Random Dynamical Systems. Lect. Not. Math. 1606, Springer-Verlag, 1995. MR 1369243 (96m:58139)

12.
Ya. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics Series, the University of Chicago Press, Chicago and London, 1997. MR 1489237 (99b:58003)

13.
M. Qian, M.-P. Qian and J.-S. Xie. Entropy formula for random dynamical systems: Relations between entropy, exponents and dimension. Ergod. Th. Dynam. Sys. 23 (2003), no. 6, 1907-1931. MR 2032494 (2004m:37096)

14.
V. A. Rokhlin. On the foundamental ideas of measure theory. Amer. Math. Soc. Transl. (1) 10 (1962), 1-52.

15.
V. A. Rokhlin. Lectures on the entropy theory of measure-preserving transformations. Russ. Math. Surveys 22:5 (1967), 1-54.

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37C45, 37H15

Retrieve articles in all Journals with MSC (2000): 37C45, 37H15

Additional Information:

Pei-Dong Liu
Affiliation: School of Mathematical Sciences, Peking University, 100871 Beijing, People's Republic of China
Email: lpd@pku.edu.cn

Jian-Sheng Xie
Affiliation: School of Mathematical Sciences, Peking University, 100871 Beijing, People's Republic of China
Address at time of publication: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing, People's Republic of China
Email: js_xie@yahoo.com.cn

DOI: 10.1090/S0002-9947-06-03933-X
PII: S 0002-9947(06)03933-X
Keywords: Random dynamical systems, Eckmann-Ruelle conjecture, hyperbolic measure
Received by editor(s): January 16, 2004
Received by editor(s) in revised form: April 11, 2004
Posted: April 11, 2006
Additional Notes: This work was supported by the 973 Fund of China for Nonlinear Science and the NSFDYS
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google