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Proceedings of the American Mathematical Society

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

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On the barycenter of the tent map
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by Kuo-Chang Chen and Xun Dong PDF
Proc. Amer. Math. Soc. 138 (2010), 4025-4035 Request permission

Abstract:

It is well known that the average position or barycenter of generic orbits for the standard tent map is $0.5$. Periodic orbits are exceptional orbits in the sense that most of them have barycenters different from $0.5$. In this paper we prove that for any positive integer $n$, there exist $n$ distinct periodic orbits for the standard tent map with the same barycenter. We also provide some patterns of periodic orbits with the same barycenter.
References
  • Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
  • M. Misiurewicz; Rotation theory. Proceedings of the RIMS Workshop on Dynamical Systems and Applications, 2006. Available at http://www.math.kyoto-u.ac.jp/$\sim$kokubu/ RIMS2006/proc.html. See also http://www.scholarpedia.org/article/Rotation_theory.
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Additional Information
  • Kuo-Chang Chen
  • Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan
  • MR Author ID: 637019
  • ORCID: 0000-0002-6618-4784
  • Email: kchen@math.nthu.edu.tw
  • Xun Dong
  • Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
  • Address at time of publication: Susquehanna International Group, 401 City Avenue, Suite 220, Bala Cynwyd, Pennsylvania 19004
  • Email: xundong@math.miami.edu
  • Received by editor(s): November 11, 2009
  • Received by editor(s) in revised form: January 19, 2010
  • Published electronically: May 17, 2010
  • Additional Notes: This work is partially supported by the National Science Council and the National Center for Theoretical Sciences (NCTS) in Taiwan. Part of the work was completed while the second author was a visiting researcher at the Center in 2007. The second author would like to thank the NCTS for its support. We also thank M. Misiurewicz for helpful comments on the first draft and the referee for helpful remarks.
  • Communicated by: Bryna Kra
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 4025-4035
  • MSC (2010): Primary 37E05; Secondary 37E45
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10397-0
  • MathSciNet review: 2679623