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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.

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