Thermodynamic formalism for intermittent maps with multiple neutral fixed points and phase transitions

In this paper, we develop the thermodynamic formalism for intermittent full-branch interval maps with finitely many neutral fixed points and for a broad class of possibly discontinuous potentials. Under a hyperbolicity assumption on the potential, we prove the existence of a unique equilibrium state with positive entropy and exponential decay of correlations. Furthermore, we characterize the phase transitions for the one-parameter family $\beta \phi$, with $\beta >0$, and prove that, at the critical parameter, multiple ergodic equilibrium states may coexist, at most one of which has positive entropy. Our approach relies on a detailed analysis of conformal measures.