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 $-t\log|Df|$, 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 $t\nearrow t^+$.