On the total variation, topological entropy and sensitivity for interval maps

Abstract

A concept related to total variation termed H 1 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 H 1 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 H 1 condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H 1 condition or sensitivity then gives the positivity of the topological entropy of f.