Write $I=[0,1]$, and let $S$ be the shift on $X=\{ (x_n)_{n\in\mathbb Z} : x_n\in I^k \}$. Is there a flow $\phi_t$ on $X$ with $\phi_1=S$? Here I require that $\phi_t$, for fixed $t$, is at least a homeomorphism on $X$, with respect to the product topology.

I'm really mainly interested in $k=2$ (though I'd also be interested in the variant where we replace the cube by a torus of that dimensions, which should make it easier to flow around). For $k=1$, the answer is *no,* for the trivial reason that $p$-periodic sequences must be invariant under $\phi_t$, so we won't be able to evolve $(\ldots 010101\ldots)$ towards its shift without crossing through a constant sequence, but these are invariant.

For $k\ge 2$, this specific problem disappears, but of course there could be other obstructions, even of this nature, since $\phi_t$ has to preserve any property that can be described dynamically in terms of $S$ (such as being an almost periodic sequence).

This sounds like the kind of question someone must have thought about already, so will perhaps be easy for the experts.