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