  Fibonacci unimodal maps can have a wild Cantor attractor, and hence be
Lebesgue dissipative, depending on the order of the critical point. We present
a one-parameter family $f_\lambda$ of countably piecewise linear unimodal
Fibonacci maps in order to study the thermodynamic formalism of dynamics where
dissipativity of Lebesgue (and conformal) measure is responsible for phase
transitions. We show that for the potential $\phi_t = -t\log|f'_\lambda|$,
there is a unique phase transition at some $t_1 \le 1$, and the pressure
$P(\phi_t)$ is analytic (with unique equilibrium state) elsewhere. The pressure
is majorised by a non-analytic $C^\infty$ curve (with all derivatives equal to
0 at $t_1 < 1$) at the emergence of a wild attractor, whereas the phase
transition at $t_1 = 1$ can be of any finite order for those $\lambda$ for
which $f_\lambda$ is Lebesgue conservative. We also obtain results on the
existence of conformal measures and equilibrium states, as well as the
hyperbolic dimension and the dimension of the basin of $\omega(c)$.
