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On homeomorphic and diffeomorphic solutions of the Abel equation on the plane. (English) Zbl 0787.39006

Let \(f\) be an orientation preserving homeomorphism of the plane onto itself with no fixed points. Then the following conditions are equivalent:
(i) \(f\) is conjugate to the translation, i.e. there exists a homeomorphism \(\varphi:\mathbb{R}^ 2 \to \mathbb{R}^ 2\) such that \(f(x)=\varphi^{-1}(\varphi(x)+a)\), \(x \in \mathbb{R}^ 2\) for an \(a \in \mathbb{R}^ 2 \backslash \{(0,0)\}\).
(ii) There exists a line \(K\) such that \(K \cap f[K]=\emptyset\), \(U \cap f[U]= \emptyset\), \(\bigcup_{n \in \mathbb{Z}}f^ n[U]=\mathbb{R}^ 2\) where \(U=M \cup f[K]\) and \(M\) is a strip bounded by \(K\) and \(f[K]\);
(iii) There exists a family of curves \(C_ \alpha\), \(\alpha \in I\) and a line \(K\) such that \(f[C_ \alpha]=C_ \alpha\), \[ \text{card }K \cap C_ \alpha=1, \] \(\alpha \in I\), \(C_ \alpha \cap C_ \beta=\emptyset\), \(\alpha \neq \beta\) and \(\bigcup_{\alpha \in T} C_ \alpha =\mathbb{R}^ 2\).
Moreover, the general continuous, homeomorphic and \(C^ p\) class solutions of the Abel equation \(\varphi (f(x))=\varphi(x)+a\) for \(x \in \mathbb{R}^ 2\) are given.
Reviewer: M.C.Zdun (Kraków)

MSC:

39B12 Iteration theory, iterative and composite equations
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable
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