Mapping Solutions in Nonmetricity Gravity: Investigating Cosmological Dynamics in Conformal Equivalent Theories

We investigate the impact of conformal transformations on the physical properties of solution trajectories in nonmetricity gravity. Specifically, we explore the phase-space and reconstruct the cosmological history of a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker universe within scalar-nonmetricity theory in both the Jordan and Einstein frames. A detailed analysis is conducted for three different connections defined in both the coincident and non-coincident gauges. Our findings reveal the existence of a unique one-to-one correspondence for equilibrium points in the two frames. Furthermore, we demonstrate that solutions describing accelerated universes remain invariant under the transformation that relates these conformally equivalent theories.