The Phase Space of $k$-Essence $f(R)$ Gravity Theory
  Among the remaining viable theories that can successfully describe the
late-time era is the $k$-Essence theory and in this work we study in detail the
phase space of $k$-Essence $f(R)$ gravity in vacuum. This theory can describe
in a viable way the inflationary era too, so we shall study the phase space in
detail, since this investigation may reveal general properties regarding the
inflationary attractors. By appropriately choosing the dimensionless variables
corresponding to the cosmological system, we shall construct an autonomous
dynamical system, and we find the fixed points of the system. We focus on
quasi-de Sitter attractors, but also to radiation and matter domination
attractors, and study their stability. As we demonstrate, the phase space is
mathematically rich since it contains stable manifold and unstable manifold.
With regard to the inflationary attractors, these exist and become
asymptotically unstable, a feature which we interpret as a strong hint that the
theory has an inherent mechanism for graceful exit from inflation. We describe
in full detail the underlying mathematical structures that control the
instability of the inflationary attractors, and also we also address the same
problem for radiation and matter domination attractors. The whole study is
performed for both canonical and phantom scalar fields, and as we demonstrate,
the canonical scalar $k$-Essence theory is structurally more appealing in
comparison to the phantom theory, a result also demonstrated in the related
literature on $k$-Essence $f(R)$ gravity.
