TY - JOUR
T1 - On the total variation, topological entropy and sensitivity for interval maps
AU - Juang, Jonq
AU - Shieh, Shih Feng
N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.
PY - 2008/5/15
Y1 - 2008/5/15
N2 - A concept related to total variation termed H1 condition was recently proposed to characterize the chaotic behavior of an interval map f by Chen, Huang and Huang [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. In this paper, we establish connections between H1 condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that H1 condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H1 condition or sensitivity then gives the positivity of the topological entropy of f.
AB - A concept related to total variation termed H1 condition was recently proposed to characterize the chaotic behavior of an interval map f by Chen, Huang and Huang [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. In this paper, we establish connections between H1 condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that H1 condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H1 condition or sensitivity then gives the positivity of the topological entropy of f.
KW - Interval maps
KW - Sensitivity
KW - Topological entropy
KW - Total variation
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U2 - 10.1016/j.jmaa.2007.11.013
DO - 10.1016/j.jmaa.2007.11.013
M3 - Article
AN - SCOPUS:38949136473
VL - 341
SP - 1055
EP - 1067
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
ER -