Actions of groups of homeomorphisms on one-manifolds (Q260098)

Let \(M\) be a connected manifold (without boundary). A homeomorphism \(f\) of \(M\) with compact support is called compactly isotopic to the identity if there exists a continuous map \(F:M \times [0, 1] \longrightarrow M\) such thatNEWLINENEWLINE\noindent (1) for any \(t_0 \in [0, 1]\), \ \(F(\cdot , t_0)\) is a homeomorphism of \(M\);NEWLINENEWLINE\noindent (2) for any \(t_0 \in [0, 1]\), \ there exists a compact subset \(K \subset M\) satisfying that \(\{ x \in M \mid F(x , t_0) \neq x \} \subset K\);NEWLINENEWLINE\noindent (3) \(F(\cdot , 0) = Id_M\) and \(F(\cdot , 1)=f\).NEWLINENEWLINEThe paper focuses on the description of all the group morphisms from the group of compactly supported homeomorphisms isotopic to the identity of a manifold, say Homeo\(_0{(M)}\), to the group of homeomorphisms of the real line or of the circle. Using his own tools, the author proves the following two theorems.NEWLINENEWLINE\smallskipNEWLINENEWLINELet \(M\) be a connected manifold of dimension greater than 2 and let \(N\) be a connected one-manifold. Then any group morphism NEWLINE\[NEWLINE\varphi: \text{Homeo}_{0}(M)\longrightarrow \text{Homeo}(N)NEWLINE\]NEWLINE is trivial.NEWLINENEWLINE\smallskipNEWLINENEWLINELet \(N\) be a connected one-manifold. For any group morphism NEWLINE\[NEWLINE \varphi:\text{Homeo}_{0}(\mathbb{R})\longrightarrow \text{Homeo}(N), NEWLINE\]NEWLINE there exists a closed set \(K \subset N\) such thatNEWLINENEWLINE\noindent (a) the set \(K\) is pointwise fixed under any homeomorphism in \(\varphi(\text{Homeo}_{0}(\mathbb{R}))\);NEWLINENEWLINE\noindent (b) for any connected component \(I\) of \(N\setminus K\), there exists a homeomorphism NEWLINE\[NEWLINE h_1:\mathbb{R} \longrightarrow I NEWLINE\]NEWLINE \noindent such that NEWLINE\[NEWLINE \varphi(f) |_{Id} = h_I f h_{I}^{-1} \;\;\;\;\;\text{ for all } \;f \in \text{Homeo}_0(\mathbb{R}). NEWLINE\]NEWLINENEWLINENEWLINECompare with the results of [\textit{K. Mann}, Ergodic Theory Dyn. Syst. 35, No. 1, 192--214 (2015; Zbl 1311.57047)] in the differentiable case.