Convergence of the local logarithmic slope Γ(R, M) under the influence of hierarchical fragmentation with the spatial scales R (solid black line) towards the Salpeter slope (dotted red line) for different initial slopes Γ₀ = −2.0, −1.5, −0.5, 0, 0.5, 1.0. This convergence is only possible if ${\xi }_M^{\prime }>{\varphi }_M^{\prime }$, with ${\xi }_M^{\prime }=\partial \xi /\partial \log M$ and ${\varphi }_M^{\prime }=\partial \varphi /\partial \log M$. The initial distribution can also converge towards other asymptotic values (dotted grey lines) depending on the relationship between ${\xi }_M^{\prime }$ and ${\varphi }_M^{\prime }$ according to Eq. (20). The slopes associated with the cases ${\xi }_M^{\prime }=0$ and ${\varphi }_M^{\prime }=0$ (dotted blue lines) correspond to the asymptotes determined in Sects. 4.2 and 4.3 respectively.