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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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