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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:
37E99 Low-dimensional dynamical systems
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