  We give the first example of a smooth family of real and complex maps having
sensitive dependence of geometric Gibbs states at positive temperature. This
family consists of quadratic-like maps that are non-uniformly hyperbolic in a
strong sense. We show that for a dense set of maps in the family the geometric
Gibbs states do not converge at positive temperature. These are the first
examples of non-convergence at positive temperature in statistical mechanics or
the thermodynamic formalism, and answers a question of van Enter and Ruszel. We
also show that this phenomenon is robust: There is an open set of analytic
2-parameter families of quadratic-like maps that exhibit sensitive dependence
of geometric Gibbs states at positive temperature.
