  We give the first example of a transitive quadratic map whose real and
complex geometric pressure functions have a high-order phase transition. In
fact, near the phase transition these functions behave as $x \mapsto \exp (- 1
/ x^2)$ near $x = 0$, before becoming linear. This quadratic map has a
non-recurrent critical point, so it is non-uniformly hyperbolic in a strong
sense.
