  For quadratic-like maps, we show a phenomenon of sensitive dependence of
geometric Gibbs states: There are analytic families of quadratic-like maps for
which an arbitrarily small perturbation of the parameter can have a definite
effect on the low-temperature geometric Gibbs states. Furthermore, 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. We introduce a geometric version of the Peierls condition for contour
models ensuring that the low-temperature geometric Gibbs states are
concentrated near the critical orbit.
