Let $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^{\infty}$ vector field. Fix a (single) real number $d$ such that
$$
1\leq d\leq n
.
$$
Under what conditions is the flow map $\Phi_V$ defined as sending any $x_0\in \mathbb{R}^n$ to the time $1$-solution of autonomous system of ODEs:
$$
\dot{x(t)} = V(x(t)) \qquad x(0)=x_0,
$$
satisfies:
$$
\mathcal{H}^d(K)= \mathcal{H}^d(\Phi(K)) \qquad \mbox{for all compact }K\subseteq \mathbb{R}^n
;
$$
where $\mathcal{H}^d$ is the $d$-dimensional Hausdorff (outer) measure on $\mathbb{R}^n$?
I'm not looking for a general characterisation (though I'd be happy to have one) but just a simple ''non-trivial'' necessary condition.
Best Answer
I would like to know why you are interested in the specific $d$-dimensional measures given by the Hausdorff ones. For $d=n$ such a choice is understandable since the Lebesgue measure in $\mathbb{R}^{n}$ is proportional to $\mathcal{H}^{n}$. But (to the best of my knowledge) there is no nice description in the case of $d$-dimensional Hausdorff measures. A difference even appears in the definition of Hausdorff measures if we change coverings by arbitrary subsets to, say, balls (we obtain the so-called spherical Hausdorff measures). I'd like to mention that
Dynamics usually have nothing to do with the original geometry of the phase space. Thus, for most purposes in dynamics it is better to let ourselves vary measures (for example, by considering different metrics in tangent spaces or adding a weight (density) function to a given measure).
To study evolution of measures via smooth dynamics (i. e. when we are armed with linearization) it is better to consider measures (or pseudo measures) or just quantities based on coverings by sets which behave well under linear transformations. But these (pseudo) measures usually provide only a two-sided majorant for the Hausdorff measures $\mathcal{H}^{d}$.
There are two standard types of nice sets: (A) parallelepipeds (which are mapped by linear transformations also to parallelepipeds) and (B) balls (which are mapped by linear transformations to ellipsoids).
(A) For the evolution of $d$-dimensional volumes of a parallelepiped with edges $\xi_{1,0},\ldots,\xi_{d,0}$ under linearization near a point $x_{0}$ over a time $t$ there is the Liouville trace formula: $$ |\xi_{1}(t) \wedge \ldots \wedge \xi_{d}(t)|_{\bigwedge^{d}\mathbb{R}^{n}} = |\xi_{1}(0) \wedge \ldots \wedge \xi_{d}(0)|_{\bigwedge^{d}\mathbb{R}^{n}} \cdot \operatorname{exp}\left( \int_{0}^{t} \operatorname{Tr} \left[ A(s) \circ \Pi(s) \right]ds \right),$$ where $A(t)=DV(x(t;x_{0}))$; $\dot{\xi}_{i}(t)=A(t)\xi_{i}(t)$ with $\xi_{i}(0)=\xi_{i,0}$; $\Pi(t)$ is the orthogonal projector onto the space spanned by $\xi_{1}(t),\ldots,\xi_{d}(t)$ and $\operatorname{Tr}$ is the trace. So, this formula shows linear evolution of natural volumes of the simplest $d$-dimensional objects. So, in some sense the condition $$\int_{0}^{t} \operatorname{Tr} \left[ A(s) \circ \Pi(s) \right]ds = 0$$ is necessary for natural $d$-dimensional volumes to be preserved under the time-$t$ map. Note that for $d=n$ there is nothing more than $\operatorname{Tr}A(s)=\operatorname{div}V(x(s,x_{0}))$ under the integral. To obtain a proper pseudo measure (for which we may have volume preserving), I may suggest to consider a $d$-dimensional pseudo measure in $\mathbb{R}^{n}$ given by coverings by $n$-dimensional parallelepipeds for which we sum their $d$-dimensional volumes given by the maximum over all $d$-dimensional volumes of its $d$-dimensional edges-parallelopipeds. The main problem to solve is to include a parallelepiped, which is perturbed by a small ball, to a larger parallelepiped, which have $d$-dimensional volume (in the introduced sense) close to the unperturbed one. I do not know if this may work and need some time to resolve it (maybe someone can do it before me).
(B) The image of the unit ball $\mathcal{B}$ under a linear transformation $L$ is an ellipsoid $\mathcal{E}$ with semi-axes $\alpha_{1} \geq \ldots \geq \alpha_{n}$, which are given by the singular values of $L$, i. e. the eigenvalues of $L^{*}L$ arranged in non-decreasing order. We can introduce a $d$-dimensional volume of the ellipsoid in the case of a non-integer $d=k+s$, where $k$ is a non-negative integer and $s \in (0,1]$, as $$\omega_{d}(\mathcal{E}):=\alpha_{1} \cdot \ldots \cdot \alpha_{k}. \alpha^{s}_{k+1}$$ Also $\omega_{d}(L):=\alpha_{1} \cdot \ldots \cdot \alpha_{k} \alpha^{s}_{k+1}$ is called the singular value function of $L$. In fact, for integer $d$ we have $$\omega_{d}(L) = \sup \frac{|L\xi_{1,0} \wedge \ldots \wedge L\xi_{d,0}|_{\bigwedge^{d}\mathbb{R}^{n}}}{|\xi_{1,0} \wedge \ldots \wedge \xi_{d,0}|_{\bigwedge^{d}\mathbb{R}^{n}}} = \sup_{|\xi_{i,0}| \leq 1, i=1,\ldots,d} |L\xi_{1,0} \wedge \ldots \wedge L\xi_{d,0}|_{\bigwedge^{d}\mathbb{R}^{n}}.$$ So, if the approach from (A) works, $\omega_{d}(D\Phi_{V})=1$ (for integer $d$) should be a natural necessary condition.