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

TL;DR

This paper demonstrates that for certain families of quadratic maps, small parameter changes can significantly alter the low-temperature geometric Gibbs states, and this sensitivity is robust across a broad class of such maps.

Contribution

It introduces a geometric Peierls condition and proves the robust sensitive dependence of Gibbs states in analytic quadratic-like map families.

Findings

Sensitive dependence of Gibbs states is shown to be robust.

An open set of quadratic-like maps exhibits this phenomenon.

A geometric Peierls condition ensures concentration near the critical orbit.

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.