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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Forcing relation on patterns of invariant sets and reductions of interval maps


Authors: Jie-Hua Mai and Song Shao
Journal: Trans. Amer. Math. Soc. 363 (2011), 2517-2549
MSC (2010): Primary 37E15, 37E05
Published electronically: December 8, 2010
MathSciNet review: 2763725
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Abstract: Patterns of invariant sets of interval maps are the equivalence classes of invariant sets under order-preserving conjugacy. In this paper we study forcing relations on patterns of invariant sets and reductions of interval maps. We show that for any interval map $ f$ and any nonempty invariant set $ S$ of $ f$ there exists a reduction $ g$ of $ f$ such that $ g\vert _S=f\vert _S$ and $ g$ is a monotonic extension of $ f\vert _S$. By means of reductions of interval maps, we obtain some general results about forcing relations between the patterns of invariant sets of interval maps, which extend known results about forcing relations between patterns of periodic orbits. We also give sufficient conditions for a general pattern to force a given minimal pattern in the sense of Bobok. Moreover, as applications, we give a new and simple proof of the converse of the Sharkovskiĭ Theorem and study fissions of periodic orbits, entropies of patterns, etc.


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Additional Information

Jie-Hua Mai
Affiliation: Institute of Mathematics, Shantou University, Shantou, Guangdong, 515063, People’s Republic of China
Email: jhmai@stu.edu.cn

Song Shao
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
Email: songshao@ustc.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05103-7
PII: S 0002-9947(2010)05103-7
Received by editor(s): October 22, 2008
Received by editor(s) in revised form: April 9, 2009
Published electronically: December 8, 2010
Additional Notes: The first author was supported by the Special Foundation of National Prior Basis Research of China (Grant No. G1999075108).
The second author was supported by National Natural Science Foundation of China (Grant No. 10871186).
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.