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$D_n\triangleq \left\{x \in \mathbb{R}^n:\, \|x\|\leq 1\right\}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function $$ \phi: [0,1]\times D_n\rightarrow D_n, $$ which is continuously differentiable in its first argument, such that the set $$ \left\{ \phi(t,x): t\in [0,1] \right\} $$ is dense in $D_n$ for some $x \in D_n$.

Are there explicit examples, in closed-form, for such a function $\phi$?

Note: If we only require $\phi$ to be continuous, then the Hahn–Mazurkiewicz guarantees the existence of such a surjective continuous function: $$ \psi:[0,1]\rightarrow D_n. $$ Taking $\phi(t,x)\triangleq \psi(t)$ gives the existence of a continuous such function. However, this result doesn't guarantee that $\phi$ is smooth or give a closed-form expression in this case...

ABIM
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  • Isn't this the same as asking for a $D_n$-filling path, by shrinking $\phi(t,D_n)$ to $\phi(t,x)$? – Ville Salo Jul 24 '20 at 13:08
  • (and by compactness one doesn't exist?) – Ville Salo Jul 24 '20 at 13:09
  • Indeed one exists by Hahn-Mazurkiewicz (I added details above) but I'm looking some something a bit more explicit (and only on a dense subset of $D_n$ really...) – ABIM Jul 24 '20 at 13:45
  • Sorry, I misread that as open disk. Obviously one exists on the closed one. – Ville Salo Jul 24 '20 at 13:51
  • Ok, based on edit history it was an open disk when I sent my comment. – Ville Salo Jul 24 '20 at 13:54
  • Actually, I only noticed the typo following your comment (so actually thanks). – ABIM Jul 24 '20 at 13:57
  • It was in parentheses because I figured it was indeed a typo, but my cellphone screen is so broken that I refigured I just misread it on the go. Also, doesn't https://mathoverflow.net/questions/201424/proof-that-no-differentiable-space-filling-curve-exists answer your question? – Ville Salo Jul 24 '20 at 14:04
  • (The image is compact so dense image is the same as surjective.) – Ville Salo Jul 24 '20 at 14:05
  • You can find tons of explicit space-filling curve constructions by a search. If you want something that "looks like a formula", see https://math.stackexchange.com/questions/921985/what-is-the-hilbert-curves-equation . (Any answer to your question with $n \geq 2$ will also give a space-filling path on $[0,1]^2$ so it seems to me that this question is a duplicate. I mean the continuous part, after ruling out smooth ones.) – Ville Salo Jul 24 '20 at 14:19

1 Answers1

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If you mean by "continuous differentiable" that for any fixed $x$, the function $t \mapsto \partial_t \phi(t,x)$ is continuous on $[0,1]$, then the image (for the same fixed $x$) of $\{\phi(t,x), t\}$ must have finite length, since $|\partial_t \phi(t,x)|$ is continuous on a closed interval and hence bounded.

But if this image is dense in $D_n$ for $n \geq 2$, it must have infinite length, a contradiction.

Proof of the latter statement: there exists a constant $c_n$ such that for every positive integer $K$, $D_n$ contains a subset $S_K$ containing $K^n$ points such that the pairwise distance between the points are at least $c_n / K$. (Just take a rectangular grid.) Therefore any space-filling curve must have length at least $c_n K^{n-1}$. Take $K\to \infty$.

Willie Wong
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