Leśniak, Zbigniew On homeomorphic and diffeomorphic solutions of the Abel equation on the plane. (English) Zbl 0787.39006 Ann. Pol. Math. 58, No. 1, 7-18 (1993). 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 Keywords:homeomorphic solutions; free mapping; diffeomorphic solutions; Abel equation PDFBibTeX XMLCite \textit{Z. Leśniak}, Ann. Pol. Math. 58, No. 1, 7--18 (1993; Zbl 0787.39006) Full Text: DOI