Random-Looking Deterministic Dynamical Systems

Question

Can we get, from a smooth vector field $V$ on $\mathbb{R}^n$ (or a suitable subset), a dynamical system which appears to be random in the following sense: let $\phi(x,t)$ be the flow of $V$ and $P$ the (non-degenerate) invariant probability measure of $V$ on $\mathbb{R}^n$. For $k\in\mathbb{R_+}$ and $m\in\mathbb{N}$, look at the set of ${(\phi(x,ak))}_{a=l}^{l+m}$ for $l\in\mathbb{N}$, and consider these as a sequence $s_{x,m,k}(l)$ indexed by $l$. Let $P_{x,m,k}(l)$ be a probability measures on ${(\mathbb{R}^n)}^{m+1}$ defined as follows by extending from open sets. Let $P_{x,m,k}(O)$ for open $O$ be equal to the natural density of the set of $l$ for which $s_{x,m,k}(l)$ belongs to $O$. Our condition is then, for all $m$, for (almost all if necessary) $x$ (under the Lebesgue measure), $P_{x,m,k}$ converges as a measure to $P^{m+1}$ in $k$, the independent product of $P$ with itself $m+1$ times. In words, if we choose large enough timesteps, the proportions given by count of times at which a trajectory and subsequent iterates (up to some limit) belong to a set are arbitrarily close to those given by randomly independently choosing the same number of points in $\mathbb{R}^n$ according to $P$. Terser, if you couldn't observe a huge number of points along the trajectory, and you couldn't observe the trajectory at very small intervals, then you wouldn't be able to say if it was random or not.