New dynamical system approach to Palatini $f(R)$ theories and its
  application to exponential gravity
  The approach of dynamical systems is a useful tool to investigate the
cosmological history that follows from modified theories of gravity. It
provides qualitative information on the typical background solutions in a
parametrized family of models, through the computation of the fixed points and
their characters (attractor, repeller or saddle), allowing, for instance, the
knowledge of which regions on the parameter space of the models generate the
desired radiation, matter and dark energy dominated eras. However, the
traditional proposal for building dynamical systems for an $f(R)$ theory in the
Palatini formalism assumes the invertibility of a function that depends on the
specific Lagrangian functional form, which is not true, for example, for the
particular theory of exponential gravity ($f(R)=R-\alpha R_*(1-e^{-R/R_*})$).
In this work, we propose an alternative choice of variables to treat $f(R)$
models in their Palatini formulation, which do include exponential gravity. We
derive some general results that can be applied to a given model of interest
and present a complete description of the phase space for exponential gravity.
We show that Palatini exponential gravity theories have a final attractor
critical point with an effective equation of state parameter $w_{\text{eff}} =
-1$ (for $\alpha>1$), $w_{\text{eff}} = -2/3$ (for $\alpha=1$) and
$w_{\text{eff}} = 0$ (for $\alpha<1$). Finally, our analytical results are
compared with numerical solutions of the field equations.
