the probability measures $d_t^{-n} (f^n)^* \eta^k$, where $d_t:= d^k$ is the topological degree of $f$ and $\eta^k$ is the standard volume form on ${\mathbb {P}}^k$. We refer to the survey article of Sibony [@Sib] for an introduction to the dynamical systems $({\mathbb {P}}^k,f,\mu)$. Fornaess-Sibony proved that $\mu$ is mixing [@FS] and Briend-Duval that $\mu$ is the unique measure of maximal entropy [@BrDu2].\
Przytycki-Urbański-Zdunik [@PUZ] introduced coding techniques for $({\mathbb {P}}^1,f,\mu)$. This allowed them to prove the Almost Sure Invariance Principle (ASIP) for Hölder and singular observables, like $\log \abs{f'}$. In the present article, we extend the coding techniques to $({\mathbb {P}}^k,f,\mu)$ and obtain the ASIP for observables which allow analytic singularities. As an application, we obtain a direct proof of the absolute continuity of $\mu$ when it satisfies Pesin’s formula. We review our results in subsections \[baba\] - \[jaja\], subsection \[relt\] is devoted to related results.

Bernoulli coding maps {#baba}
---------------------

Let us endow $\Sigma
:=  \{ 1, \ldots, d_t \}^{\mathbb {N}}$ with the natural product measure $\nu:= \otimes_{n = 0}^\infty \bar
\nu$, where $\bar \nu$ is equidistributed on $\{ 1, \ldots, d_t \}$. We denote by ${{\tilde \alpha}}$ the elements of $\Sigma$ and by $s$ the left shift acting 