the root of the Brownian map (i.e. $A^+(\epsilon)$): the previous bound was $\epsilon^{4-\delta}$ for $\delta>0$ arbitrary small, we obtain $\epsilon^{4}|\log(\epsilon)|^{1+\delta}$. We moreover obtain the lower bound $\epsilon^4|\log(\epsilon)|^{-q}$ for any $q>6$, thanks to Proposition \[Proposition lower envelope A(t)\]. Again, the analogue holds for stable maps.

The organisation of the paper is the following. In Section 2, we formally introduce the growth-fragmentations setting. This includes definitions of positive self-similar Markov processes and Lamperti’s transformation, the intrinsic area measure of a growth-fragmentation, as well as the two spinal decompositions introduced in [@BBCK16] that are central throughout this work. We also establish an important Markov-branching property of $A$ in Lemma \[Lemma Markov-branching property of A\], which, roughly speaking, reduces the study of $A$ to that of a single self-similar Markov process.

We state our results in Section 3. Theorem \[Theorem area GF\] concerns the almost sure behaviour of $A(\epsilon)$ as $\epsilon\to 0+$, when $\mathbf{X}$ starts from a typical cell with positive initial size. Our second result is Proposition \[Theorem upper envelope A(t)\] which provides an upper bound for the almost sure behaviour of $A^+(\epsilon)$ when the initial cell starts from $0$ and behaves differently from the others, with indefinite growth. A lower bound is 