the whole cell-system indexed by $\mathbb{U}$, and we denote $\mathcal{P}_1$ its law. More generally, we can construct such a cell-system with the Eve cell starting from a size $x>0$, its law is then denoted $\mathcal{P}_x$.

#### The growth-fragmentation.

The growth-fragmentation process $(\mathbf{X}(t))_{t\geq 0}$ induced by the cell-system is the process following the collection of particles’ sizes in time, forgetting about their genealogy, that is $$\begin{aligned}
        \mathbf{X}(t):=\left\{\!\!\left\{\chi_u(t-b_u):u\in\mathbb{U},b_u\leq t<b_u+\zeta_u\right\}\!\!\right\},
    \end{aligned}$$ where the elements are repeated according to their multiplicity.

The law of $\mathbf{X}$ is characterized by $\alpha$ and a particular function $\kappa:\mathbb{R}_+\to\mathbb{R}\cup\{\infty\}$ called the *cumulant function* of $\mathbf{X}$ (see [@S17]), which is defined as $$\begin{aligned}
        \kappa(q):=\psi(q)+\int_{{\left(-\infty\,,0\right)}}(1-e^y)^q\Lambda(\mathrm{d}y).
    \end{aligned}$$ The importance of $\kappa$ for the study of self-similar growth-fragmentations comes from the fact, taken from [@B17] Lemma 3, that $\mathbb{E}_x\left(\sum_{i\geq 1}\chi_i(0)^{q}\right)=x(1-\kappa(q)/\psi(q))$, when it makes sense. This means that when $\kappa(q)=0$, the initial size of a cell is equal to the expectation of the sum of the sizes raised to the power $q$ of its first generation children at their birthtime. For this reason, we assume throughout this work that the *Cramér hypothesis* holds, that is there exists $\omega_->0$ such that $$\begin{aligned}
\label{Equation Cramer hypothesis}
        \kappa(\omega_-)=0,\quad-\infty<\kappa'(\omega_-)<0.
    \end{aligned}$$ We suppose that $\kappa$ has a second root 