holds for every $A \in {\mathcal{R}}$. If, in addition, $X$ is compact Hausdorff and $\mu$ is a Borel measure then there exists a residual subset ${\mathcal{R}}'\subset C^0(X,{\mathbb{G}})$ such that measurable dichotomy holds for every $A \in {\mathcal{R}}'$.

The proposition follows from the results of [@B] mentioned in remark \[r.when A varies\], see appendix \[ss.easydich\] for details.

\[q.connectedness\] Can theorem \[t.measclassif\] be extended in some form to non-connected manifolds? Does corollary \[c.measdich\] remain true if $X$ is not connected?

The continuous case
-------------------

Next we consider volume-preserving homeomorphisms as dynamics. As explained before, it is useful to assume differentiability of the map $A$.

So now we let $X$ be a smooth compact connected manifold, possibly with boundary, of dimension $d \ge 2$, and let $\mu$ be a smooth volume measure. Recall ${\mathit{Homeo}}(X,\mu)$ indicates the set of $\mu$-preserving homeomorphisms, endowed with the $C^0$ topology. Our main result in that setting is:

\[t.cont\] Let $T\in{\mathit{Homeo}}(X,\mu)$ and $A \in C^1(X, {\mathbb{G}})$. Assume there is a $T$-periodic point $p = T^n(p)$ such that:

-   $A_T^n(p)$ is elliptic;

-   $A$ is not locally constant at at least one of the points $p$, $Tp$, …, $T^{n-1}p$.

Then for every ${\varepsilon}>0$ there exists $\tilde{T} \in {\mathit{Homeo}}(X,\mu)$ arbitrarily $C^0$-close 