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A structure theorem in one dimensional dynamics. (English) Zbl 0737.58020
Let $${\mathcal A}$$ be the class of smooth maps $$f: I\to I$$ such that $$f(0)=f(1)=0$$ and $$f$$ has a unique critical point $$c\in (0,1)$$. Then it has been shown that the logistic map $$f_ \rho(x)=\rho x(1-x)$$ has a structural importance in $${\mathcal A}$$ analogous to that of the irrational rotation $$R_ \mu$$ through the angle $$\mu$$ for circle homeomorphisms $$g: S^ 1\to S^ 1$$ with rotation number $$\mu$$. More specifically, the irrational rotation $$R_ \mu$$ gives an ordering of the orbits of $$g$$ even when the map $$g$$ is not topologically conjugate to $$R_ \mu$$. The analogous property for $$f\in {\mathcal A}$$ is that there exists a $$\rho\in [0,4]$$ such that $$\{c\}\cup\{f^{-1}(c)\}\cup\cdots\cup\{f^{-n}(c)\}$$ and $$\{{1\over 2}\}\cup\{f_ \rho^{-1}({1\over 2})\}\cup\cdots\cup\{f_ \rho^{-n}({1\over 2})\}$$ are in the same order in $$[0,1]$$. This enables a semi-conjugacy $$h$$ to be constructed such that $$h f=f_ \rho h$$ and gives an analogue of Denjoy’s theory for maps of $${\mathcal A}$$. If $$L=h^{-1}(x)$$ then the interval $$L$$ is either periodic or wandering.
Main theorem: If the critical point of $$f\in {\mathcal A}$$ is nonflat (i.e. $$D^ kf(c)\neq 0$$ for some $$k$$) then $$f$$ has no wandering interval.

##### MSC:
 3.7e+100 Low-dimensional dynamical systems
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