  With any max-stable random process $\eta$ on $\mathcal{X}=\mathbb{Z}^d$ or
$\mathbb{R}^d$, we associate a random tessellation of the parameter space
$\mathcal{X}$. The construction relies on the Poisson point process
representation of the max-stable process $\eta$ which is seen as the pointwise
maximum of a random collection of functions $\Phi=\{\phi\_i, i\geq 1\}$. The
tessellation is constructed as follows: two points $x,y\in \mathcal{X}$ are in
the same cell if and only if there exists a function $\phi\in\Phi$ that
realizes the maximum $\eta$ at both points $x$ and $y$, i.e. $\phi(x)=\eta(x)$
and $\phi(y)=\eta(y)$. We characterize the distribution of cells in terms of
coverage and inclusion probabilities. Most interesting is the stationary case
where the asymptotic properties of the cells are strongly related to the
ergodic properties of the non-singular flow generating the max-stable process.
For example, we show that: i) the cells are bounded almost surely if and only
if $\eta$ is generated by a dissipative flow, ii) the cells have positive
asymptotic density almost surely if and only if $\eta$ is generated by a
positive flow.
