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

Free energy and equilibrium states for families of interval maps