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We analyze the global behaviour of the growth index of cosmic inhomogeneitiesin an isotropic homogeneous universe filled by cold non-relativistic matter anddark energy (DE) with an arbitrary equation of state. Using a dynamical systemapproach, we find the critical points of the system. That unique trajectory forwhich the growth index $\gamma$ is finite from the asymptotic past to theasymptotic future is identified as the so-called heteroclinic orbit connectingthe critical points $(\Omega_m=0,~\gamma_{\infty})$ in the future and$(\Omega_m=1,~\gamma_{-\infty})$ in the past. The first is an attractor whilethe second is a saddle point, confirming our earlier results. Further, in thecase when a fraction of matter (or DE tracking matter) $\varepsilon \Omega^{\rmtot}_m$ remains unclustered, we find that the limit of the growth index in thepast $\gamma_{-\infty}^{\varepsilon}$ does not depend on the equation of stateof DE, in sharp contrast with the case $\varepsilon=0$ (for which$\gamma_{-\infty}$ is obtained). We show indeed that there is a mathematicaldiscontinuity: one cannot obtain $\gamma_{-\infty}$ by taking$\lim_{\varepsilon \to 0} \gamma^{\varepsilon}_{-\infty}$ (i.e. the limits$\varepsilon\to 0$ and $\Omega^{\rm tot}_m\to 1$ do not commute). We recover inour analysis that the value $\gamma_{-\infty}^{\varepsilon}$ corresponds totracking DE in the asymptotic past with constant$\gamma=\gamma_{-\infty}^{\varepsilon}$ found earlier.

Global properties of the growth index: mathematical aspects and physical relevance