Free energy and equilibrium states for families of interval maps
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 (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.
Citation
Dobbs , N & Todd , M J 2023 , ' Free energy and equilibrium states for families of interval maps ' , Memoirs of the American Mathematical Society , vol. 286 , no. 1417 . https://doi.org/10.1090/memo/1417
Publication
Memoirs of the American Mathematical Society
Status
Peer reviewed
ISSN
0065-9266Type
Journal article
Rights
Copyright © 2020 American Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://www.ams.org/publications/ebooks/memoirs
Description
Funding: MT was partially supported by FCT grant SFRH/BPD/26521/2006 and NSF grants DMS0606343 and DMS 0908093. ND was supported by ERC Bridges project, the Academy of Finland CoE in Analysis and Dynamics Research and an IBM Goldstine fellowship.Collections
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