[@CL] a quantified version of the Briend-Duval theorem. Our version is similar, but we shall give a different proof.\
The systems $({\mathbb {P}}^k,f,\mu)$ whose measure $\mu$ is absolutely continuous with respect to Lebesgue measure were characterized by Berteloot, Dupont and Loeb [@BeDu], [@BL]. In that case, $f$ is semi-conjugated to an affine dilation on a complex torus, these maps are the so-called *Lattès examples*. We note that theorem D characterizes these maps by the minimality of the Lyapunov exponents. Another characterization of Lattès examples involves the *Hausdorff dimension* of $\mu$, defined as the infimum of the Hausdorff dimension of Borel sets with full $\mu$-measure (see Pesin’s book [@P2]): Dinh-Dupont [@DD] proved that $\dim_\HH(\mu) = 2k$ if and only if the exponents are minimal. In the context of $({\mathbb {P}}^1,f,\mu)$, Mañé proved that $\log d = \dim_\HH(\mu) \cdot \chi$, where $\chi$ denotes the Lyapunov exponent of $\mu$. In particular, the function $L:= \log d
- \dim_\HH(\mu) \cdot \log \abs{f'}$ is a $\mu$-centered observable. Zdunik [@Z] proved that ${\sigma}_L = 0$ if and only if $f$ is a Lattès example, a Tchebychev polynomial or a power $z^{\pm d}$. The proof relies on the classification of critically finite fractions with parabolic Thurston’s orbifold.

