\, , \, f^n(x) = x \}$, this set is at most countable. Let $\CC$ be the critical set of $f$, $\VV:= \cup_{i = 0}^\infty
f^i(\CC)$ and $\VV_n:= \cup_{i = 1}^ n f^i(\CC)$. The degree of $\VV_n$, denoted $\tau_n$, is equal to $(d + \ldots +
d^n) \deg \CC$ counted with multiplicity.\
The equilibrium measure $\mu$ is defined as the limit of $\mu_{n,z}:=
{1 \over d_t ^n} \sum_{f^n(y) = z} \delta_y$, where $\delta_y$ denotes the Dirac mass at $y$. In that definition, $z$ has to be taken outside a totally invariant algebraic set $\EE \subset \VV$, the so-called exceptional set of $f$. We denote by $\JJ$ the support of $\mu$. The measure $\mu$ is mixing and satisfies $\mu(f(B)) = d_t
\, \mu(B)$ whenever $f$ is injective on $B$. It is the unique measure of maximal entropy (equal to $\log d_t$). The Lyapunov exponents $\chi_1 \leq \ldots \leq \chi_k$ of $\mu$ are larger than or equal to $\log
d^{1/2}$. They satisfy the classical formula $\int_{{\mathbb {P}}^k} \log {{\textsf {Jac} \,  }}f \, d\mu = 2 (
\chi_1 + \ldots +\chi_k )$, where ${{\textsf {Jac} \,  }}f$ is the non-negative $\CC^\infty$ function on ${\mathbb {P}}^k$ satisfying $f^* \eta^k = {{\textsf {Jac} \,  