We study for the first time the dynamical properties and the growth index of linear matter perturbations of the Finsler-Randers (FR) cosmological model, for which we consider that the cosmic fluid contains matter, radiation and a scalar field. Initially, for various FR scenarios we implement a critical point analysis and we find solutions which provide cosmic acceleration and under certain circumstances we can have de-Sitter points as stable late-time attractors. Then we derive the growth index of matter fluctuations in various Finsler-Randers cosmologies. Considering cold dark matter and neglecting the scalar field component from the perturbation analysis we find that the asymptotic value of the growth index is $gamma_{infty}^{(FR)}approxfrac {9}{16}$, which is close to that of the concordance $Lambda$ cosmology, $gamma^{(Lambda)} approxfrac{6}{11}$. In this context, we show that the current FR model provides the same Hubble expansion with that of Dvali, Gabadadze and Porrati (DGP) gravity model. However, the two models can be distinguished at the perturbation level since the growth index of FR model is $sim18.2%$ lower than that of the DPG gravity $gamma^{(DGP)} approx frac{11}{16}$. If we allow pressure in the matter fluid then we obtain $gamma_{infty}^{(FR)}approxfrac{9(1+w_{m})(1+2w_{m})}{2[8+3w_{m}% (5+3w_{m})]}$, where $w_{m}$ is the matter equation of state parameter. Finally, we extend the growth index analysis by using the scalar field and we find that the evolution of the growth index in FR cosmologies is affected by the presence of scalar field.
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