that $\mu$ is a Monge-Ampère mass. In particular, the product of the dilation rates satisfies ${{\textsf {Jac} \,  }}f^n(x) \geq \norm{(d_x f^n)^{-1}}^{-2k}  \geq (d^{n/2}/n)^{2k} = d^{kn}/n^{2k}$. Now if we assume ${\sigma}_J > 0$, then the function $\log {{\textsf {Jac} \,  }}f^n$ would present non trivial oscillations around its mean value $\log d^{kn}$, due to the CLT. More precisely, it would imply $\log {{\textsf {Jac} \,  }}f^n \leq \log d^{kn} - {\sigma}_J \sqrt n$ on a subset of $\mu$-measure $\simeq \int_{-\infty}^{-1}
e^{-u^2/2}$. That contradicts the preceding estimate, hence ${\sigma}_J  = 0$. We deduce the absolute continuity of $\mu$ from the cocycle property $J = u - u
\circ f$ $\mu$-a.e. and a linearization property of the dynamics along typical negative orbits [@BeDu].

Related results {#relt}
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The systems $({\mathbb {P}}^k , f , \mu)$ and $(\Sigma , s
, \nu)$ are actually conjugated by a bimeasurable map up to zero measure subsets, that property was proved by Briend [@Br]. However, the regularity of the conjugacy seems difficult to handle. Let us also mention that finite-to-one coding maps $({\mathbb {P}}^k,f,\mu) \to (\Sigma,s,\nu)$ were constructed by Buzzi [@Bu] by means of suitable partitions of ${\mathbb {P}}^k$.\
The ASIP has been proved for many 