The information topology and true laminations for diffeomorphisms

Author:
Meiyu Su

Journal:
Conform. Geom. Dyn. **8** (2004), 36-51

MSC (2000):
Primary 37D30; Secondary 37C05

DOI:
https://doi.org/10.1090/S1088-4173-04-00107-9

Published electronically:
March 8, 2004

MathSciNet review:
2060377

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We explore the lamination structure from data supplied by a general measure space provided with a Borel probability measure . We show that if the data satisfy some typical axioms, then there exists a lamination injected in the underlying space whose image fills up the measure . For an arbitrary -diffeomorphism of a compact Riemannian manifold , we construct the data that naturally possess the properties of the axioms; thus we obtain the stable and unstable laminations continuously injected in the stable and unstable partitions . These laminations intersect at almost every regular point for the measure.

**1.**L. Barreire, Ya. Pesin and J. Schmeling,*Dimension and Product Structure of Hyperbolic Measures*, Annals of Math.**(149)**(1999), 755-783. MR**2000f:37027****2.**J. P. Eckman and D. Ruelle,*Ergodic Theory of Chaos and Strange Attractors*, Rev. Modern Phys.**(57)**(1885), 617-656. MR**87d:58083a****3.**A. Fathi, M.-R. Herman and J.-C. Yoccoz,*A proof of Pesin's stable manifold theorem*, Geometric Dynamics (J. Palis, Jr., Ed.), Lecture Notes in Math., Vol. 1007, Springer-Verlag, (1983), 177-215. MR**85j:58122****4.**F. Ledrappier and L. S. Young,*The Metric Entropy of Diffeomorphisms, Part I: Characterization of Measures Satisfying Pesins Entropy Formula, and Part II: Relations between Entropy, Exponents and Dimension,*Annals of Math.**(122)**(1985), 509-539 and 540-574.**5.**R. Mañè,*A proof of Pesin's formula*, Ergodic Theory and Dynamical Systems**(1)**(1981), 95-102. MR**83b:58042****6.**V. I. Oseledec,*Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems*, Trans. Mosc. Math. Soc.**(19)**(198), 197-221.**7.**Ya. B. Pesin,*Families of invariant manifolds corresponding to non-zero characteristic exponents*, English Transl., Math USSR-Izv.**(10)**(1976), 1261-1305.**8.**-,*Characteristic Lyapunov exponents and smooth ergodic theory*, English transl., Russian Math. Surveys**(32)**(1977), 55-114.**9.**C. Pugh and M. Shub,*Ergodic Attractors*, Trans. of AMS, Vol. 312**(1)**(1989), 1-54. MR**90h:58057****10.**G. E. Shilov and B. L. Gurevich, Translated by R. A. Silverman,*Integral, measure and derivative: a unified approach*, Dover Publications, INC. (1977), 209-215. MR**57:6342****11.**M. Su,*Measured solenoidal Riemann surfaces and holomorphic dynamics*, Journal of Differential Geometry**(47)**(1997), 170-195. MR**99a:58133****12.**-,*Laminations for hyperbolic measures*, Preprint, May 2003.**13.**M. Su and D. Sullivan,*Laminations for endmorphisms*, in preparation, 2003.**14.**D. Sullivan,*Bounds, quadratic Differentials, and Renormalization Conjectures*, Math. into the Twenty-first Century**(Vol. 2)**, Providence, RI, AMS (1991). MR**93k:58194****15.**-,*Linking The Universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers: topological methods in modern mathematics*, Publish or Perish Inc. (1993). MR**94c:58060**

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Additional Information

**Meiyu Su**

Affiliation:
Mathematics Department, Long Island University, Brooklyn Campus, 1 University Plaza, Brooklyn, New York 11201

Email:
msu@liu.edu

DOI:
https://doi.org/10.1090/S1088-4173-04-00107-9

Keywords:
$C^{1 +\alpha}$-diffeomorphisms on Riemannian manifolds,
stable and unstable manifolds and partitions,
laminations,
Pesin boxes,
and information topology

Received by editor(s):
September 10, 2003

Received by editor(s) in revised form:
January 29, 2004

Published electronically:
March 8, 2004

Article copyright:
© Copyright 2004
American Mathematical Society