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)



Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps

Authors: Henk Bruin and Stefano Luzzatto
Journal: Proc. Amer. Math. Soc. 134 (2006), 265-272
MSC (2000): Primary 37B10; Secondary 37A35, 11K99, 37A45
Published electronically: August 19, 2005
MathSciNet review: 2170567
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.

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

  • 1. A. Blokh, Decomposition of dynamical systems on an interval, Russian Math. Surveys 38 (1983) 133-134. MR 0718829 (86d:54060)
  • 2. H. Bruin, The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps, Ergod. Th. and Dynam. Sys. 18 (1998) 555-565. MR 1631716 (99i:58090)
  • 3. H. Bruin, G. Keller, Equilibrium states for $S$-unimodal maps, Ergod. Th. and Dynam. Sys. 18 (1998) 765-789. MR 1645373 (2000g:37039)
  • 4. Yongluo Cao, Stefano Luzzatto, Isabel Rios, Minimum principle for Lyapunov exponents and a higher-dimensional version of a Theorem of Mañé, to appear in Qual. Th. of Dyn. Sys.
  • 5. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. of Math. 34 (1979) 213-237. MR 0570882 (82c:28039a)
  • 6. G. Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989) 183-200. MR 1026617 (91b:28011)
  • 7. G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergod. Th. and Dynam. Sys. 10 (1990) 717-744. MR 1091423 (92e:58118)
  • 8. O. Kozlovski, Getting rid of the negative Schwarzian derivative condition, Ann. Math. 152 (2000) 743-762. MR 1815700 (2002e:37050)
  • 9. F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergod. Th. and Dynam. Sys. 1 (1981) 77-93. MR 0627788 (82k:28018)
  • 10. S. Luzzatto, L. Wang, Topological invariance of generic non-uniformly expanding multimodal maps, Preprint (2003).
  • 11. M. Martens, W. de Melo, S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992) 273-318. MR 1161268 (93d:58137)
  • 12. W. de Melo, S. van Strien, One-Dimensional Dynamics, Springer, Berlin, Heidelberg, New York (1993). MR 1239171 (95a:58035)
  • 13. T. Nowicki, F. Przytycki, Topological invariance of the Collet-Eckmann property for $S$-unimodal maps, Fund. Math. 155 (1998) 33-43. MR 1487986 (99a:58058)
  • 14. T. Nowicki, D. Sands, Non-uniform hyperbolicity and universal bounds for $S$-unimodal maps, Invent. Math. 132 (1998) 633-680. MR 1625708 (99c:58122)
  • 15. F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993) 309-317. MR 1186141 (93k:58193)
  • 16. F. Przytycki, J. Rivera-Letelier, S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003) 29-63. MR 1943741 (2003k:37065)
  • 17. S. van Strien, E. Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc. 17 (2004) 749-782. MR 2083467
  • 18. M. Todd, One-dimensional dynamics: cross-ratios, negative Schwarzian and structural stability, Ph.D. thesis Warwick (2004).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37B10, 37A35, 11K99, 37A45

Retrieve articles in all journals with MSC (2000): 37B10, 37A35, 11K99, 37A45

Additional Information

Henk Bruin
Affiliation: Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom

Stefano Luzzatto
Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom

Received by editor(s): September 5, 2004
Published electronically: August 19, 2005
Additional Notes: The authors thank Juan Rivera-Letelier for drawing their attention to previous results relating to Lemma 1 and Proposition 1. They also thank Feliks Przytycki for pointing out an error in an earlier version of this paper.
Communicated by: Michael Handel
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society