Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps

Abstract

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.

The title changed. A discussion about the topology and combinatorics of the maps was added before the Main Theorem

References in corpus (4)

• Chaotic temperature dependence at zero temperature
• Zero-temperature phase diagram for double-well type potentials in the summable variation class
• Sensitive dependence of geometric Gibbs states at positive temperature
• Complex maps without invariant densities