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Roots of continuous piecewise monotone maps of an interval. (English) Zbl 0736.58026
A map $$f: I\rightarrow I$$ of a closed interval $$I$$ is said to have an iterative root of $$n$$-th order, if there is a map $$g: I\rightarrow I$$ such that $$f=g^ n$$, the exponential denoting $$n$$-fold composition of functions. The paper studies the existence and properties of iterative roots of continuous, piecewise monotone or piecewise linear maps.
The main results are for short: Continuous, piecewise monotone maps, which have continuous roots of order $$n$$, also have piecewise monotone roots of this order. If additionally the map is surjective, then every root is piecewise monotone. The authors show that a similar theorem for piecewise linear maps does not hold. The latter is shown by studying in detail the properties of roots of piecewise linear horseshoe maps. The produced counterexample also provides an answer for a problem posed by K. Simon at the third Czechoslovak Summer School on Dynamical Systems, which was the starting point for this paper.
Reviewer: C.H.Cap (Zürich)

##### MSC:
 37B99 Topological dynamics 37E99 Low-dimensional dynamical systems 26A18 Iteration of real functions in one variable 54H20 Topological dynamics (MSC2010)
##### Keywords:
iterative roots; horseshoe maps
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