zbMATH — the first resource for mathematics

On maximal parallelizable regions of flows of the plane. (English) Zbl 1106.39023
Let \(\{f^t: t \in \mathbb R\}\) be a flow of free mappings (i.e., for every \(t\in \mathbb R \setminus \{0\}\) the function \(f^t\) is a homomorphism of \(\mathbb R^2\) onto itself without fixed points and which preserves orientation). An invariant region \(M \subset \mathbb R^2\) is said to be parallelizable if there exists a homomorphism \(\varphi\) mapping \(M\) onto \(\mathbb R^2\) such that \(f^t(x)=\varphi^{-1}(\varphi(x)+(t,0))\) for \(x \in M\). The following equivalence relation can be defined: \(p \sim q\) if \(p=q\) or \(p\) and \(q\) are endpoints of some arc \(K\) for which \(f^n(K)\to \infty\) as \( n \to \pm\infty\). The main result of the paper says that a maximal parallelizable region \(M\) of \(\{f^t: t \in \mathbb R\}\) is a union of equivalence classes.

39B12 Iteration theory, iterative and composite equations
54H20 Topological dynamics (MSC2010)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces