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Explicit transitive flow on disc

MathOverflow Asked on December 25, 2021

$D_ntriangleq left{x in mathbb{R}^n:, |x|leq 1right}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function
$$
phi: [0,1]times D_nrightarrow D_n,
$$

which is continuously differentiable in its first argument, such that the set
$$
left{
phi(t,x): tin [0,1]
right}
$$

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…

One Answer

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 $Kto infty$.

Answered by Willie Wong on December 25, 2021

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