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<p>For a piecewise monotone function <i>F</i> of height 1, an open question was raised: Does <i>F</i> have an iterative root <i>f</i> of order <i>n</i> ≤ <i>N</i>(<i>F</i>) + 1 if the ‘characteristic endpoints condition’ is not satisfied? This question was answered partly in the case that <i>F</i> is strictly increasing on its characteristic interval <i>K</i>(<i>F</i>) but <i>f</i> is strictly decreasing on <i>K</i>(<i>F</i>). In this paper we discuss the question for <i>F</i> increasing on <i>K</i>(<i>F</i>) in some remaining cases, giving the necessary and sufficient conditions for the existence of continuous iterative roots <i>f</i> decreasing on <i>K</i>(<i>F</i>) of order <i>n</i> = <i>N</i>(<i>F</i>) &gt; 2 with <i>H</i>(<i>f</i>) = <i>n</i> − 1.</p>

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A Note on Characteristic Endpoints Question for Decreasing Iterative Roots on Characteristic Interval

  • 期刊论文

  • 发表日期：

    2025年6月15日

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  • Siyi Zhao,
  • Liu Liu

摘要

For a piecewise monotone function F of height 1, an open question was raised: Does F have an iterative root f of order n ≤ N(F) + 1 if the ‘characteristic endpoints condition’ is not satisfied? This question was answered partly in the case that F is strictly increasing on its characteristic interval K(F) but f is strictly decreasing on K(F). In this paper we discuss the question for F increasing on K(F) in some remaining cases, giving the necessary and sufficient conditions for the existence of continuous iterative roots f decreasing on K(F) of order n = N(F) > 2 with H(f) = n − 1.