The set $\DD$ satisfies ${{\rm Vol \, }}(\DD) = 0$.

The proof is postponed to subsection \[volo\]. We now state the quantified version. The constant $\tilde c$ has been introduced in the statement of Briend-Duval’s theorem, and we denote by $L$ the complex line containing $z$ and $w$.

\[good\] There exists ${\epsilon}> 0$ such that for every distinct points $(z,w) \notin \DD \cup \VV$, there exist an injective smooth path ${\gamma}:
[0,1] \to L \setminus \VV$ joining $z$ and $w$, a decreasing family of topological discs $(\Delta_n)_n \subset L$ and an integer $n_{z,w}$ such that for any $n \geq n_{z,w}$:

1.  ${\gamma}[0,1] \subset \Delta_n \subset L \setminus \VV_{\theta_n}$,

2.  there exist $(1-d^{-\theta n }) d_t ^n$ inverse branches of $f^n$ on $\Delta_n$,

3.  these branches satisfy ${{\textsf {diam} \, }}g_n (\Delta_n)  \leq \tilde c \,  d^{- {\epsilon}n }$.

We note that $\theta,{\epsilon}$ and $\tilde c$ do not depend on $(z,w) \in {\mathbb {P}}^k \setminus (\DD \cup \VV)$.

Construction of good paths in the complex line $L \subset {\mathbb {P}}^k$ {#ppaa}
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Let $(z,w)$ be distinct points in ${\mathbb {P}}^k \setminus (\DD \cup \VV
)$. We identify the complex line $L$ containing $z$ and $w$ with the $2$-dimensional sphere. We recall 