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\(PM\) functions, their characteristic intervals and iterative roots. (English) Zbl 0873.39009
Let \(N(F)\) denote the number of non-monotone points of a continuous strictly piecewise monotone (PM) function \(F:I= [a,b]\to I\) and let \(H(F)= \min\{k\geq 1: N(F^k)= N(F^{k+1})\}\), where \(F^1=F\) and \(F^{k+1}=F\circ F^k\). An interval \([a',b']\subset I\) is called the characteristic interval of \(F\) iff
(i) \(a'\) and \(b'\) are either non-monotone points of \(F\) or endpoints of \(I\),
(ii) there are no non-monotone points inside \((a',b')\),
(iii) \([m,M]\subset [a',b']\), where \(m=\inf F\), \(M=\sup F\).
The author proved that \(F\) has no continuous iterative roots of order \(n\) for \(n>N(F)\) in the case \(H(F)>1\). Furthermore, the author establishes the relations between the existence of continuous iterative roots and the behaviour of \(F\) with \(H(F)\leq 1\), on its characteristic interval.

39A12 Discrete version of topics in analysis
39B22 Functional equations for real functions
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