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Product Type Potential on the One-Dimensional Lattice Systems: Selection of Maximizing Probability and a Large Deviation Principle

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Abstract

Given an uncountable and compact metric space E, the one-dimensional lattice system is the space \(\Omega =E^\mathbb {N}\) with an a priori measure p on the state space E. Given a potential \(f: \Omega \rightarrow \mathbb {R}\) one can ask: among the invariant probabilities which one is the equilibrium probability \(\mu \) for the interaction described by f? As usual the equilibrium probability for f is the one maximizing pressure. We will present here the case of the product type potential on the one-dimensional lattice system and in this setting we can show the explicit expression of the equilibrium probability. We will also consider questions about Ergodic Optimization, maximizing probabilities, subactions and we will show selection of a maximizing probability, when temperature goes to zero. Finally we show a large deviation principle when temperature goes to zero and we present an explicit expression for the deviation function.

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Notes

  1. \(f(\beta )\sim g(\beta )\) as \(\beta \rightarrow \infty \) if \(\lim _{\beta \rightarrow \infty }\frac{f(\beta )}{g(\beta )}=1.\)

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Acknowledgements

We thanks Leandro Cioletti and Artur Lopes for their suggestions and comments on the early version of this manuscript.

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  1. IME, UFRGS, Porto Alegre, Brazil

    J. Mohr

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  1. J. Mohr
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J. Mohr is partially supported by CNPq.   Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Mohr, J. Product Type Potential on the One-Dimensional Lattice Systems: Selection of Maximizing Probability and a Large Deviation Principle. Qual. Theory Dyn. Syst. 21, 44 (2022). https://doi.org/10.1007/s12346-022-00576-z

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