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Additive, Almost Additive and Asymptotically Additive Potential Sequences Are Equivalent

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Abstract

Motivated by various applications and examples, the standard notion of potential for dynamical systems has been generalized to almost additive and asymptotically additive potential sequences, and the corresponding thermodynamic formalism, dimension theory and large deviations theory have been extensively studied in the recent years. In this paper, we show that every such potential sequence is actually equivalent to a standard (additive) potential in the sense that there exists a continuous potential with the same topological pressure, equilibrium states, variational principle, weak Gibbs measures, level sets (and irregular set) for the Lyapunov exponent and large deviations properties. In this sense, our result shows that almost and asymptotically additive potential sequences do not extend the scope of the theory compared to standard potentials, and that many results in the literature about such sequences can be recovered as immediate consequences of their counterpart in the additive case. A corollary of our main result is that all quasi-Bernoulli measures are weak Gibbs.

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Notes

  1. If \({{\mathcal {F}}}_*(\mu )\) can take the value \(-\infty \), which is not the case with almost and asymptotically additive sequences, a more detailed statement is required in order to remove some ambiguity, see for example [18, Theorem 1.1].

  2. Our main result does not apply to general sub-additive sequences and we refer the reader to [18] and [10, Chapters II.4 and III.7] for a detailed exposition of the sub-additive thermodynamic formalism and its applications. The same comment applies to asymptotically sub-additive sequences, see for example [27, 55].

  3. Such measures were considered in [37, Definition 8.2], see also [22, 44].

  4. In practice, it is most common to require the functions \(f_n\) to be at least measurable, but this is not necessary for Theorem 4.6.

  5. The quantity \(\Vert f_n - S_n f\Vert _\infty \) in (4.8) is allowed to be infinite, since both f and \(f_n\) may be unbounded.

  6. There are minor variations of this condition in the literature, compare for example with equation (15) in [13] and equation (2.2) in [14].

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Acknowledgements

I would like to thank V. Jakšić and A. Shirikyan for providing a clever improvement in the proof of the main result, and C.-E. Pfister for suggesting a reformulation that led to Theorem 2.1 below. I am also very thankful to N. Dobbs, J.-P. Eckmann, B. Fernandez and C.-A. Pillet for stimulating conversations and useful comments about the manuscript.

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  1. Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université de Paris – Sorbonne Université – CNRS, 75205, Paris Cedex 13, France

    Noé Cuneo

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Cuneo, N. Additive, Almost Additive and Asymptotically Additive Potential Sequences Are Equivalent. Commun. Math. Phys. 377, 2579–2595 (2020). https://doi.org/10.1007/s00220-020-03780-7

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