on $\Sigma$. Let $\JJ$ be the support of $\mu$. The following theorem yields coding maps ${\omega}: \Sigma \to \JJ$ up to zero measure sets. The set $\SS \subset {\mathbb {P}}^k$ will be defined in section \[GCT\], it has zero Lebesgue measure.

[A:]{} Let $z \in {\mathbb {P}}^k \setminus \SS$. There exist an $s$-invariant set $\Sigma'
\subset \Sigma$ of full $\nu$-measure and an $f$-invariant set $\JJ'
\subset \JJ$ of full $\mu$-measure satisfying the following properties. For any ${{\tilde \alpha}}\in
\Sigma'$, the point ${\omega}({{\tilde \alpha}}):= \lim_{n \to \infty} z_n({{\tilde \alpha}}) \in \JJ'$ is well defined. We have ${\omega}_* \nu  = \mu$ and the following diagram commutes: $$\xymatrix{
           \Sigma'  \ar[rr]^{s}  \ar[d]_{\omega}&    & \Sigma'  \ar[d]^{\omega}\\
           \JJ'  \ar[rr]^{f}       & &  \JJ'        }$$ Moreover there exist $\theta , {\epsilon}> 0$, $n_z \geq 1$ and $\tilde n : \Sigma' \to {\mathbb {N}}$ larger than $n_z$ such that:

1.  $d(z_n({{\tilde \alpha}}),{\omega}({{\tilde \alpha}}) ) \leq \tilde c_{\epsilon}\, d^{-{\epsilon}n}$ for every ${{\tilde \alpha}}\in \Sigma'$ and $n \geq \tilde n ({{\tilde \alpha}})$,

2.  $\nu ( \{ \tilde n  \leq q \} )  \geq 1 - c_\theta \, d^{-\theta q}$ for every $q \geq n_z$.

We note that $\Sigma'$, $\JJ'$ and ${\omega}$ depend on $z \in {\mathbb {P}}^k \setminus \SS$, but ${\omega}_* 