## Large deviation principles for countable Markov shifts

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## Abstract:

We establish the large deviation principle for a topological Markov shift over infinite alphabet which satisfies strong connectivity assumptions called “finite irreducibility” or “finite primitiveness”. More precisely, we assume the existence of a Gibbs state for a potential $\phi$ in the sense of Bowen, and prove the level-$2$ large deviation principles for the distribution of empirical means under the Gibbs state, as well as that of weighted periodic points and iterated preimages. The rate function is written with the pressure and the free energy associated with the potential $\phi$.## References

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## Additional Information

**Hiroki Takahasi**- Affiliation: Keio Institute of Pure and Applied Sciences, Department of Mathematics, Keio University, Yokohama 223-8522, Japan
- MR Author ID: 790386
- Email: hiroki@math.keio.ac.jp
- Received by editor(s): July 23, 2018
- Received by editor(s) in revised form: February 6, 2019, and February 12, 2019
- Published electronically: July 1, 2019
- Additional Notes: This research was partially supported by the Grant-in-Aid for Young Scientists (A) of the JSPS 15H05435, the Grant-in-Aid for Scientific Research (B) of the JSPS 16KT0021, and the JSPS Core-to-Core Program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7831-7855 - MSC (2010): Primary 37A45, 37A50, 37A60, 60F10
- DOI: https://doi.org/10.1090/tran/7829
- MathSciNet review: 4029683