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Level-2 Large Deviation Principle for Countable Markov Shifts Without Gibbs States

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Abstract

We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen’s Gibbs state. To deal with non-compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations. Under this condition we establish the level-2 Large Deviation Principle. We apply our results to the continued fraction expansion of real numbers in [0, 1) generated by the Rényi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the Rényi map.

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  1. In fact, one can show \(P(\Phi _{\beta ,P(\beta \phi )})=0\). See [23] for example.

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Acknowledgements

The author thanks the referees for their careful readings of the manuscript and giving useful suggestions for improvements. This research was partially supported by the JSPS KAKENHI 19K21835 and 20H01811.

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  1. Department of Mathematics, Keio Institute of Pure and Applied Sciences (KiPAS), Keio University, Yokohama, 223-8522, Japan

    Hiroki Takahasi

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Communicated by Marco Lenci.

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Takahasi, H. Level-2 Large Deviation Principle for Countable Markov Shifts Without Gibbs States. J Stat Phys 190, 120 (2023). https://doi.org/10.1007/s10955-023-03126-2

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