Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ satisfying:

  • $\forall t \in \mathbb{R}, \Phi(t,\cdot) \in \text{Homeo}(M)$
  • $\forall t,s \in \mathbb{R}, \Phi(t+s,\cdot) = \Phi(t,\Phi(s,\cdot))$

We consider the following $\mathcal{C}^0$ metric on continuous flows: $$\delta(\Phi,\Psi) = \sup_{t \in [0,1],x \in M} d(\Phi(t,x),\Psi(t,x)) $$

Question: Is it possible to approximate a $\mathcal{C}^0$ flow by a $\mathcal{C}^1$ flow (or even $\mathcal{C}^{\infty}$) in the $\mathcal{C}^0$ topology ? In other words, given a $\varepsilon > 0$, is there a $\mathcal{C}^1$ flow $\Psi$ such that $\delta(\Phi,\Psi) < \varepsilon$ ?

I know (from http://arxiv.org/abs/0901.1002) that this result is far from being trivial for homeomorphisms. It is true in dimension $\leq 3$ (any homeomorphism can be uniformly approximated by a diffeomorphism) and in dimension $\geq 5$ if and only if the homeo is isotopic to a diffeo. Apparently, it is still open in dimension $4$. In particular, this shows, that any element $\Phi^t$ of a continuous flow can be individually uniformly approximated by a diffeo, but this doesn't answer the question.

I have tried to look up for a reference in literature, but I couldn't find any. Has anyone any idea and/or reference on the question ?


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