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When lower entropy implies stronger Devaney chaos
Author(s):
Grzegorz
Haranczyk;
Dominik
Kwietniak
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2063-2073.
MSC (2000):
Primary 37B40, 37B20;
Secondary 37E05, 37E10
Posted:
December 15, 2008
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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).
References:
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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:
10.1090/S0002-9939-08-09756-6
PII:
S 0002-9939(08)09756-6
Received by editor(s):
August 18, 2008
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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