Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 1055-1067 |
| Number of pages | 13 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 341 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2008 May 15 |
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Keywords
- Interval maps
- Sensitivity
- Topological entropy
- Total variation
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
Cite this
On the total variation, topological entropy and sensitivity for interval maps. / Juang, Jonq; Shieh, Shih Feng.
In: Journal of Mathematical Analysis and Applications, Vol. 341, No. 2, 15.05.2008, p. 1055-1067.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On the total variation, topological entropy and sensitivity for interval maps
AU - Juang, Jonq
AU - Shieh, Shih Feng
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
UR - http://www.scopus.com/inward/record.url?scp=38949136473&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=38949136473&partnerID=8YFLogxK
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 -