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Free Energy and Equilibrium States for Families of Interval Maps
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eBook
Free Energy and Equilibrium States for Families of Interval Maps
2023
Overview
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 (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper
semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one
varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential
tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of
correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.
Publisher
American Mathematical Society
Subject
ISBN
9781470461263, 1470461269
