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Shadowing, entropy and a homeomorphism of the pseudoarc


Authors: Piotr Koscielniak and Piotr Oprocha
Journal: Proc. Amer. Math. Soc. 138 (2010), 1047-1057
MSC (2000): Primary 37B45; Secondary 54H20, 37B40, 37B05
DOI: https://doi.org/10.1090/S0002-9939-09-10162-4
Published electronically: November 10, 2009
MathSciNet review: 2566570
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Abstract: In this article we provide a method of constructing continuous maps $ f\colon [0,1]\rightarrow [0,1]$ such that $ f$ is topologically mixing, has the shadowing property, and the inverse limit of copies of $ [0,1]$ with $ f$ as the bonding map is the pseudoarc. Such a map can be obtained as an arbitrarily small $ \mathcal{C}^0$-perturbation of any topologically exact map on $ [0,1]$. We have therefore answered, in the affirmative, a question posed by Chen and Li in 1993.


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Additional Information

Piotr Koscielniak
Affiliation: Institute of Mathematics of the Jagiellonian University, ul. Lojasiewicza 6, 30-348 Kraków, Poland
Email: piotr.koscielniak@im.uj.edu.pl

Piotr Oprocha
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain – and – Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: oprocha@agh.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-09-10162-4
Keywords: Pseudoarc, shadowing, entropy, crooked map, topological mixing
Received by editor(s): May 4, 2009
Received by editor(s) in revised form: August 4, 2009
Published electronically: November 10, 2009
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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