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A Convex Analysis Approach to Entropy Functions, Variational Principles and Equilibrium States

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Abstract

Using methods from Convex Analysis, for each generalized pressure function we define an upper semi-continuous affine entropy-like map, establish an abstract variational principle for both countably and finitely additive probability measures and prove that equilibrium states always exist. We show that this conceptual approach imparts a new insight on dynamical systems without a measure with maximal entropy, may be used to detect second-order phase transitions, prompts the study of finitely additive ground states for non-uniformly hyperbolic transformations and grants the existence of finitely additive Lyapunov equilibrium states for singular value potentials generated by linear cocycles over continuous self-maps.

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Acknowledgements

The authors are grateful to the anonymous referees for the apposite comments and valuable suggestions that have helped us to improve the manuscript. The authors also thank G. Iommi, B. Kloeckner, A. Lopes, J. Rivera-Letelier and M. Todd for their insightful remarks on the first version of this text. AB was partially supported by the inner Lodz University Grant 11/IDUB/DOS/2021. MC, MM and PV were partially supported by CMUP, which is financed by national funds through FCT—Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. MC also acknowledges financial support from the project PTDC/MAT-PUR/4048/2021. PV benefited from the Grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call, awarded by FCT, and from the project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017).

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Łódź University, Lodz, Poland

    Andrzej Biś

  2. CMUP & Faculdade de Ciências da Universidade do Porto, Porto, Portugal

    Maria Carvalho

  3. CMUP & Faculdade de Engenharia da Universidade do Porto, Porto, Portugal

    Miguel Mendes

  4. CMUP & Departamento de Matemática, Universidade Federal da Bahia, Salvador, Brazil

    Paulo Varandas

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  1. Andrzej Biś
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  2. Maria Carvalho
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Correspondence to Paulo Varandas.

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Communicated by C. Liverani.

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Biś, A., Carvalho, M., Mendes, M. et al. A Convex Analysis Approach to Entropy Functions, Variational Principles and Equilibrium States. Commun. Math. Phys. 394, 215–256 (2022). https://doi.org/10.1007/s00220-022-04403-z

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