Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents


Authors: Stefano Luzzatto and Fernando J. Sánchez-Salas
Journal: Proc. Amer. Math. Soc. 141 (2013), 3157-3169
MSC (2010): Primary 37D25
DOI: https://doi.org/10.1090/S0002-9939-2013-11565-0
Published electronically: May 24, 2013
MathSciNet review: 3068969
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that for any $ C^{1+\alpha } $ diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence $ \Omega _{n} $ of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence $ \{\mu _{n}\} $ of $ f $-invariant ergodic probability measures with $ supp (\mu _{n}) \subseteq \Omega _{n} $ we have $ \mu _{n}\to \mu $ in the weak-$ * $ topology.


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

  • 1. L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115. Cambridge University Press, 2007. MR 2348606 (2010c:37067)
  • 2. X. Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok. Nonlinearity, 23 (2010) 513-528. MR 2586367
  • 3. D. Dolgopyat and Y. B. Pesin, Every compact manifold carries a completely hyperbolic diffeomorphism. Ergodic Theory and Dynamical Systems, 22(2) (2002) 409-435. MR 1898798 (2003b:37050)
  • 4. K. Gelfert, Repellers for non-uniformly expanding maps with singular or critical points, Bulletin of the Brazilian Mathematical Society, 41(2) (2010) 237-257. MR 2738913
  • 5. B. Hasselblat, Hyperbolic Dynamical Systems, Handbook of dynamical systems, Vol. 1A, 239-319, North-Holland, Amsterdam, 2002. MR 1928520 (2004b:37047)
  • 6. B. Hasselblat and A. Katok, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, Cambridge, 1995. MR 1326374 (96c:58055)
  • 7. M. Hirayama, Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems. Series A, 9(5) (2003) 1185-1192. MR 1974422 (2004a:37032)
  • 8. A. Katok, Lyapunov exponents, entropy and periodic orbits of diffeomorphisms. Publ. Math. IHES, 51 (1980) 137-173. MR 573822 (81i:28022)
  • 9. A. Katok and L. Mendoza, Dynamical systems with non-uniformly hyperbolic behavior. Supplement to Introduction to the Modern Theory of Dynamical Systems, by A. Katok and B. Hasselblat, Encyclopedia of Math. and its Applications, Vol. 54. Cambridge University Press, 1995. MR 1326374 (96c:58055)
  • 10. C. Liang, G. Liu, and W. Sun, Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems. Transactions of the American Mathematical Society, 361(3) (2009) 1543-1579. MR 2457408 (2009m:37085)
  • 11. L. Mendoza, Ergodic attractors for diffeomorphisms of surfaces. Journal of the London Math. Soc., 37 (1988) 362-374. MR 928529 (89e:58071)
  • 12. S. Newhouse, Lectures on Dynamical Systems, Dynamical Systems (C.I.M.E. Summer School Bressanone, 1978), pp. 1-114, Progr. Math., 8, Birkhäuser, Boston, Mass., 1980. MR 589590 (81m:58028)
  • 13. Ya. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Math. USSR Izv., 40(6) (1976) 1261-1305. MR 0458490 (56:16690)
  • 14. C. Robinson, Dynamical Systems: Stability, symbolic dynamics and chaos. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396532 (97e:58064)
  • 15. Fernando J. Sánchez-Salas, Ergodic attractors as limits of hyperbolic horseshoes, Ergodic Theory and Dynamical Systems, 22 (2002) 571-589. MR 1898806 (2003c:37037)
  • 16. I. Ugarcovici, On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems. Series A, 16(2) (2006) 505-512. MR 2226494 (2007c:37025)
  • 17. Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Transactions of the American Mathematical Society, 362(8) (2010) 4267-4282. MR 2608406 (2011m:37030)
  • 18. S. Wiggins, Global bifurcations and chaos: analytical methods, Applied Mathematical Sciences, Vol. 73, Springer, New York, 1988. MR 956468 (89m:58057)
  • 19. C. Wolf and K. Gelfert, On the distribution of periodic orbits. Discrete and Continuous Dynamical Systems, Series A 26(3) (2010) 949-966. MR 2600724 (2011g:37080)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37D25

Retrieve articles in all journals with MSC (2010): 37D25


Additional Information

Stefano Luzzatto
Affiliation: Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
Email: luzzatto@ictp.it

Fernando J. Sánchez-Salas
Affiliation: Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad del Zulia, Avenida Universidad, Edificio Grano de Oro, Maracaibo, Venezuela
Email: fjss@fec.luz.edu.ve

DOI: https://doi.org/10.1090/S0002-9939-2013-11565-0
Keywords: Non-uniformly hyperbolic systems, uniformly hyperbolic systems, approximation of hyperbolic measures
Received by editor(s): June 7, 2011
Received by editor(s) in revised form: July 14, 2011, and November 25, 2011
Published electronically: May 24, 2013
Additional Notes: Most of this work was carried out at the Abdus Salam International Centre for Theoretical Physics (ICTP). The second author was partially supported by the Associateship Programme of ICTP
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society