of ${\mathbb {P}}^k$.

The exponential decay of correlations ensures the convergence of $(\star)$. This was proved in the context of $({\mathbb {P}}^k,f,\mu)$ by Fornaess-Sibony [@FS] for $C^2$ observables and by Dinh-Sibony for Hölder observables [@DS2]. Dinh-Nguyen-Sibony have recently extended that property for differences of quasi-plurisubharmonic functions (the so-called *dsh* functions) [@DNS2]. The proof relies on exponential estimates for plurisubharmonic functions with respect to $\mu$. They also obtained in that article a Large Deviations Theorem for bounded dsh and Hölder observables. In [@DNS1], Dinh-Nguyen-Sibony proved the local CLT for $({\mathbb {P}}^1,f,\mu)$ by using the theory of perturbed operators.

Denker-Przytycki-Urbański [@DPU] employed a geometric method to prove the convergence of $(\star)$ for $({\mathbb {P}}^1,f,\mu)$ and Hölder observables. The idea was to compare $\Lambda^n \psi(z)$ to $\Lambda^n
\psi(z')$ by using the contraction of most of the inverse branches of $f^n$. The cornerstone is a precise analysis of the dynamics near the critical points in the support of $\mu$. Cantat-Leborgne [@CL] extended this approach to $({\mathbb {P}}^k,f,\mu)$. A crucial ingredient was a polynomial estimate for the $\mu$-measure of postcritical neighbourhoods (lemma 5.7 of [@CL]). The original proof of that lemma contains a gap, the authors have recently proposed another one. Cantat-Leborgne also established in 