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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.

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