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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

When lower entropy implies stronger Devaney chaos


Authors: Grzegorz Haranczyk and Dominik Kwietniak
Journal: Proc. Amer. Math. Soc. 137 (2009), 2063-2073
MSC (2000): Primary 37B40, 37B20; Secondary 37E05, 37E10
Published electronically: December 15, 2008
MathSciNet review: 2480288
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Abstract: It is proved that the infimum of the topological entropy of continuous topologically exact interval (circle) maps is strictly smaller than the infimum of the topological entropy of continuous interval (circle) maps, which are topologically mixing, but not exact. Interpreting this result in terms of popular notions of chaos, one may say that on the interval (circle) lower entropy implies stronger Devaney chaos. Moreover, the infimum of the entropy of mixing circle maps is computed. These theorems may be considered as a completion of some results of Alsedà, Kolyada, Llibre, and Snoha (1999).


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Additional Information

Grzegorz Haranczyk
Affiliation: Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Email: gharanczyk@gmail.com

Dominik Kwietniak
Affiliation: Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Email: dominik.kwietniak@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09756-6
PII: S 0002-9939(08)09756-6
Received by editor(s): August 18, 2008
Published electronically: December 15, 2008
Additional Notes: The second author was supported in part by the Ministry of Science and Education grant no. N 201 2723 33 for the years 2007–2009.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.