Free energy and equilibrium states for families of interval maps

TL;DR

This paper investigates the thermodynamic properties of one-dimensional dynamical systems, focusing on the continuity of free energy and equilibrium states, and provides conditions for their stability or instability.

Contribution

It establishes almost upper-semicontinuity of free energy for families of interval maps and explores conditions for the existence and continuity of equilibrium states.

Findings

Free energy is almost upper-semicontinuous under general conditions.

Counterexamples show instability of equilibrium states without strong hypotheses.

Results contribute to understanding statistical stability in dynamical systems.

Abstract

We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (statistical stability). Counterexamples to statistical stability in the absence of strong hypotheses are provided.