Equilibrium states at freezing phase transition in unimodal maps with flat critical point

An $S$-unimodal map $f$ with flat critical point satisfying the Misiurewicz condition displays a freezing phase transition in positive spectrum. We analyze statistical properties of the equilibrium state ${\mu }_t$ for the potential $-t\log \mathrm{\mid }Df\mathrm{\mid }$, as well as how the phase transition slows down the rate of decay of correlations. We show that ${\mu }_t$ has exponential decay of correlations for all inverse temperature $t$ contained in the positive entropy phase $(t^{-},t^{+})$. If the critical point is not too flat, then the freezing point $t^{+}$ is equal to $1$, and the absolutely continuous invariant probability measure (acip for short) is the unique equilibrium state at the transition. We exhibit a case in which the acip has sub-exponential decay of correlations and ${\mu }_t$ converges weakly to the acip as $t\nearrow t^{+}$.