Global properties of the growth index: mathematical aspects and physical
  relevance
  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
$\varepsilon\to 0$ and $\Omega^{\rm tot}_m\to 1$ 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.
