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| Let $\phi:\mathbb{R}\times M\to M$ be a non-singular continuous dynamical system defined on a differentiable $n$-manifold $M$. If $\phi$ is generated by a $C^1$ vector field, then $M$ can be covered with flow boxes $D^{n-1}\times I$ on which $\phi$ is a local flow along the $I$-fibers and $D^{n-1}$ is the $(n-1)$ disk. This in general is not the case for $n >3$. In 1957, R.H.~Bing gave an example of a generalized 3-manifold, which is not homeomorphic to $\mathbb{R}^3$, but its product with $\mathbb{R}^1$ is
homeomorphic to $\mathbb{R}^4$. Thus there is a $C^0$ dynamical system defined on $\mathbb{R}^4$ not admitting flow boxes.
We consider the feasibility of defining generalized flow boxes and their applications. |
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