Generalities {#cons}
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The holomorphic systems $({\mathbb {P}}^k,f,\mu)$ {#consa}
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We introduce in this section the systems $({\mathbb {P}}^k,f,\mu)$. We refer to the articles [@BrDu1], [@BrDu2], [@FS] and [@Sib] for definitions and properties. Here ${\mathbb {P}}^k$ denotes the complex projective space of dimension $k$. We denote by $\eta$ the Fubini-Study form on ${\mathbb {P}}^k$. This is a $(1,1)$-form defined in homogeneous coordinates by ${i \over 2 \pi} \partial  \bar \partial \log \norm{z}^2$. It induces the standard metric on ${\mathbb {P}}^k$, the volume of ${\mathbb {P}}^k$ with respect to this metric is equal to $1$. The form $\eta$ induces on every complex line $L \subset {\mathbb {P}}^k$ the spherical metric with area $1$. Let $f$ be an holomorphic endomorphism of ${\mathbb {P}}^k$ with algebraic degree $d \geq 2$. It is defined in homogeneous coordinates by $[ P_0 :
  \cdots : P_k ]$ where the $P_i$ are homogeneous polynomials of degree $d$ (without common zero except the origin). The topological degree of $f$ is $d_t:= d^k$. An inverse branch of $f^n$ on $U \subset {\mathbb {P}}^k$ is an injective holomorphic map $g_n$ satisfying $f^n \circ g_n = {{\rm Id}}_U$. We let ${{\textsf {Per}  }}f:= \cup_{n \geq 1} \{ x \in {\mathbb {P}}^k
