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Because flowmaps are homeomorphic maps on a compact domain $\Omega$, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system?

that is, for any $\varepsilon>0$ does there exist $\dot z(t)=v(z(t))$ and end time $\tau>0$ with initial value $z(0)=x$ such that $\Phi(x)=\phi_v^\tau(x)$ or $\|\Phi(x)-\phi_v^\tau(x)\|<\varepsilon$ where $\phi_v^\tau(x)=z(\tau)$?

There is another version (time-fixed) of this question in Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

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  • $\begingroup$ Flow maps are not necessarily diffeomorphisms; this is true only for compact manifolds while $R^n$ is not compact. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 19, 2023 at 15:43

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Any orientation-preserving diffeomorphism of $\mathbb{R}^d$ is isotopic to the identity. Differentiating the isotopy gives you a flow whose time-one is the diffeomorphism. See also this MO answer.

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