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Monotone maps of $ G$-like continua with positive topological entropy yield indecomposability


Author: Hisao Kato
Journal: Proc. Amer. Math. Soc. 147 (2019), 4363-4370
MSC (2010): Primary 37B45, 37B40; Secondary 54H20, 54F15
DOI: https://doi.org/10.1090/proc/14602
Published electronically: April 18, 2019
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Abstract: In the previous paper Adv. Math. 304 (2017), pp. 793-808, we proved that if for any graph $ G$, a homeomorphism on a $ G$-like continuum $ X$ has positive topological entropy, then the continuum $ X$ contains an indecomposable subcontinuum. Also, if for a tree $ G$, a monotone map on a $ G$-like continuum $ X$ has positive topological entropy, then the continuum $ X$ contains an indecomposable subcontinuum. In this note, we extend these results. In fact, we prove that if for any graph $ G$, a monotone map on a $ G$-like continuum $ X$ has positive topological entropy, then the continuum $ X$ contains an indecomposable subcontinuum. Also we study topological entropy of monotone maps on Suslinean continua.


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Additional Information

Hisao Kato
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki, 305-8571 Japan
Email: hkato@math.tsukuba.ac.jp

DOI: https://doi.org/10.1090/proc/14602
Keywords: Topological entropy, continuum, indecomposable, monotone map, inverse limit, $G$-like continuum
Received by editor(s): September 21, 2016
Received by editor(s) in revised form: January 14, 2019
Published electronically: April 18, 2019
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society