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Suppose I have a map $f:X\to X$ on some reasonable space $X$. (I'm particularly thinking about a smooth map on a compact manifold or a homeomorphism of a compact CW-complex, but feel free to work in whatever category is convenient.) How can I determine whether there exists a flow $f_t$ on $X$ with $f_1 = f$; that is, a function $f_t(x):\mathbb{R}^{\geq 0} \times X \to X$ (in the same smoothness category as $f$) with $f_t(f_{t'}) = f_{t+t'}$? Obviously I need $f$ to be homotopic to the identity; is that sufficent?

anomaly
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    In the smooth category the answer is no: see e.g. this question. You can think about this in terms of Lie groups: you're asking whether the exponential map of the diffeomorphism group is surjective. Since this is already false for some matrix groups, it shouldn't be too surprising it's false for something as complicated as big and complicated as a diffeomorphism group. In more positive news, it's true that flows generate the connected component of the identity under composition. – Anthony Carapetis Jan 28 '18 at 02:57

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