On the total variation, topological entropy and sensitivity for interval maps |
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Authors: | Jonq Juang |
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Institution: | a Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, ROC b Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC |
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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. |
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Keywords: | Interval maps Total variation Topological entropy Sensitivity |
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