\nu = \mu$ holds true for any such $z$. Observe also that ${\omega}$ is not necessarily injective. The proof of theorem A (see section \[GCT\]) is based on the construction of a geometric coding tree, following the approach of Przytycki-Urbański-Zdunik [@PUZ] for $({\mathbb {P}}^1,f,\mu)$. The point $z$ is the *root* of the tree, and the set $\{ z_n({{\tilde \alpha}}) , {{\tilde \alpha}}\in \Sigma \}$ is a suitable enumeration of the $d_t^{n+1}$ points of $f^{-(n+1)}(z)$, these are *vertices* of the tree. The convergence of $(z_n({{\tilde \alpha}}))_n$ for a generic ${{\tilde \alpha}}\in \Sigma$ is obtained by constructing $d_t$ good paths joining $z$ to $w \in f^{-1}(z)$, whose inverse images decrease exponentially. In the context of $({\mathbb {P}}^1,f,\mu)$, that property was obtained in [@PUZ] by using Koebe distortion theorem. The difficulty in higher dimensions is to substitute this argument. We establish for that purpose a quantified version of a theorem of Briend-Duval (see section \[BD\]).

The class $\UU$ and approximation by cylinders {#byby}
----------------------------------------------

[:]{} An observable $\psi : {\mathbb {P}}^k \to {\mathbb {R}}\cup \{ -
  \infty \}$ belongs to the class $\UU$ if:

- $e^\psi$ is $h$-Hölder for some $h > 0$,

- $\NN_\psi:= \{ \psi = -\infty \}$ is a (possibly 