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On indecomposability in chaotic attractors


Authors: Jan P. Boroński and Piotr Oprocha
Journal: Proc. Amer. Math. Soc. 143 (2015), 3659-3670
MSC (2010): Primary 54F15
DOI: https://doi.org/10.1090/S0002-9939-2015-12526-9
Published electronically: March 25, 2015
MathSciNet review: 3348807
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Abstract: We exhibit a Li-Yorke chaotic interval map $ F$ such that the inverse limit $ X_F=\varprojlim \{F,[0,1]\}$ does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if $ \varphi $ has positive entropy then $ X_\varphi $ contains an indecomposable subcontinuum. Each subcontinuum of $ X_F$ is homeomorphic to one of the following: an arc, or $ X_F$, or a topological ray limiting to $ X_F$. Through our research, we found that it follows that $ X_F$ is a chaotic attractor of a planar homeomorphism. In addition, $ F$ can be modified to give a cofrontier that is a chaotic attractor of a planar homeomorphism but contains no indecomposable subcontinuum. Finally, $ F$ can be modified, without removing or introducing new periods, to give a chaotic zero entropy interval map, such that the corresponding inverse limit contains the pseudoarc.


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Additional Information

Jan P. Boroński
Affiliation: National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic — and — AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: jan.boronski@osu.cz

Piotr Oprocha
Affiliation: AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland — and — Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
Email: oprocha@agh.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2015-12526-9
Keywords: Interval map, arc-like, Li-Yorke chaotic, indecomposable continuum
Received by editor(s): March 29, 2013
Received by editor(s) in revised form: February 14, 2014
Published electronically: March 25, 2015
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society