Dynamical system analysis of the cosmological phases in Palatini $k$-essence gravity

We formulate a generalized $k$-essence model in the presence of a Palatini $f(\mathcal{R})$ gravitational sector. In the corresponding biscalar-tensor theory, we discuss the distinguished dynamical properties of the two scalar fields, elucidating how the Palatini scalaron can be still algebraically solved in terms of matter, the $k$-essence field and its kinetic term. We derive the conditions ensuring the absence of Ostrogradsky modes and the well-posedness of the initial data problem, also providing an intriguing analogy with a specific class of DHOST theories. Then, we investigate the cosmology of a flat Friedmann-Lema\^{i}tre-Robertson-Walker spacetime according a dynamical system approach, with the aim of determining the set of fixed points in the phase space, representing specific periods of the Universe evolution and characterized by different effective barotropic index $w_{\text{eff}}$. The analysis reveals the presence of a range of possible configurations, with the existence of (quasi) de-Sitter epochs connected by heteroclinic orbits, scaling solutions and quintessence phases.