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I have a very general question.

If I have a smooth map $\phi:X\to X$ with $X$ compact, what kind of strategics should I try to prove that $\phi$ is the time 1 of a flow in $X$? Any information or reference will be helpful. If there are well known facts but that requires more structure I'll be grateful make me note it.

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1 Answer 1

Let me give you some examples of how the problem behaves when we tweak the hypothesis. You can check the link below and the references within for further information.

First, if $X$ is a connected subset of $\mathbb{R}$ and $\phi$ is a homeomorphism of $X$ then $\phi$ can be embedded in a flow if, and only if, $\phi$ is order-preserving. Now we have examples of both flowable and un-flowable applications.

Unless inherent to your specific problem, there is no a priori need for $X$ to be compact. The case $X = \mathbb{R}^2$ is very interesting, and here it will make a big difference if $\phi$ has fixed points. You'll find bellow some positive results and a beautiful theorem by S. Andrea that rules out some possibilities.

If your question is more about smooth phenomena, you might want to consider diffeomorphisms on manifolds. Results by Palis ensures that diffeomorphisms that embeed to flows are very rare. Nevertheless, some positive results are known.

An overview of the subject and my source: "The embedding of homeomorphisms in continuous flows" by W. R. Utz online at http://www.topology.auburn.edu/tp/reprints/v06/

(this paper is from 1981, and there are also more recent results you will be able to find using some of the keywords above).

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