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Large Deviation Principle of Nonconventional Ergodic Averages

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Abstract

This paper establishes the large deviation principle (LDP) of certain types of nonconventional ergodic averages, namely, \(\frac{1}{N}S_N^*\) and \(\frac{1}{N}S_N^\#\) on \(\mathbb {N}\) (defined later). The LDP for both averages are presented and such a result extends the preceding work of (Carinci et al. in Indag Math 23(3):589–602, 2012) to some specific cases of d-multiple averages for \(d\ge 3\).

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References

  1. Ban, J.C., Hu, W.G., Lai, G.Y.: On the entropy of multidimensional multiplicative integer subshifts. J. Stat. Phys. 182(2), 1–20 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Ban, J.C., Hu, W.G., Lai, G.Y.: Thermodynamic formalism and large deviation principle of multiplicative Ising models. http://arxiv.org/abs/2203.08970

  3. Ban, J.C., Hu, W.G., Lin, S.S.: Pattern generation problems arising in multiplicative integer systems. Ergod. Theory Dyn. Syst. 39(5), 1234–1260 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ban, J.C., Lin, S.S.: Patterns generation and transition matrices in multi-dimensional lattice models. Discret. Contin. Dyn. Syst. 13(3), 637–658 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ban, J.C., Hu, W.G., Lin, S.S., Lin, Y.H.: Zeta functions for two-dimensional shifts of finite type. Am. Math. Soc. 221, 1037 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Ban, J.C., Hu, W.G., Lin, S.S., Lin, Y.H.: Verification of mixing properties in two-dimensional shifts of finite type. J. Math. Phys. 62, 072703 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Ban, J.C., Lin, S.S., Lin, Y.H.: Patterns generation and spatial entropy in two dimensional lattice models. Asian J. Math. 11(3), 497–534 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourgain, J.: Double recurrence and almost sure convergence. J. für die reine und angewandte Mathematik 404, 140–161 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Carinci, G., Chazottes, J.R., Giardina, C., Redig, F.: Nonconventional averages along arithmetic progressions and lattice spin systems. Indag. Math. 23(3), 589–602 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chazottes, J.R., Redig, F.: Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems. Electron. J. Probab. 19, 1–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dembo, A., Zeitouni, O.: LDP for Finite Dimensional Spaces. Large Deviations Techniques and Applications, pp. 11–70. Springer, New York (2009)

    Book  Google Scholar 

  12. Fan, A.H.: Some Aspects of Multifractal Analysis. Geometry and Analysis of Fractals, pp. 115–145. Springer, New York (2014)

    Book  MATH  Google Scholar 

  13. Fan, A.H., Liao, L.M., Ma, J.H.: Level sets of multiple ergodic averages. Monatshefte für Mathematik 168(1), 17–26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan, A.H., Schmeling, J., Wu, M.: Multifractal analysis of some multiple ergodic averages. Adv. Math. 295, 271–333 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Host, B., Kra, B.: Nonconventional ergodic averages and nilmanifolds. Ann. Math. 161, 397–488 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Theory Dyn. Syst. 32(5), 1567–1584 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frantzikinakis, N.: Some open problems on multiple ergodic averages. Bull. Hell. Math. Soc. 60, 41–90 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding, 2nd edn. Cambridge University Press, Cambridge (2021)

    Book  MATH  Google Scholar 

  19. Peres, Y., Schmeling, J., Seuret, S., Solomyak, B.: Dimensions of some fractals defined via the semigroup generated by 2 and 3. Isr. J. Math. 199(2), 687–709 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Peres, Y., Solomyak, B.: Dimension spectrum for a nonconventional ergodic average. Real Anal. Exch. 37(2), 375–388 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Winkler, M.: The algorithmic structure of the finite stopping time behavior of the 3x+1 function. http://arxiv.org/abs/1709.03385 (2017)

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Acknowledgements

We would like to sincerely thank the anonymous referee for providing inspiring comments and helpful suggestions for the first draft of this article. These significantly improve the readability and solidify the validity of theorems in the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, National Chengchi University, Taipei, 11605, Taiwan, Republic of China

    Jung-Chao Ban

  2. Math. Division, National Center for Theoretical Science, National Taiwan University, Taipei, 10617, Taiwan, Republic of China

    Jung-Chao Ban

  3. College of Mathematics, Sichuan University, Chengdu, 610064, China

    Wen-Guei Hu

  4. Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, 30010, Taiwan, Republic of China

    Guan-Yu Lai

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  1. Jung-Chao Ban
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  2. Wen-Guei Hu
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Correspondence to Guan-Yu Lai.

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Communicated by Yoshiko Ogata.

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Ban is partially supported by the Ministry of Science and Technology, ROC (Contract MOST 109-2115-M-004-002-MY2 and 108-2115-M-004-003). Hu is partially supported by the National Natural Science Foundation of China (Grant 11601355).

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Ban, JC., Hu, WG. & Lai, GY. Large Deviation Principle of Nonconventional Ergodic Averages. J Stat Phys 190, 61 (2023). https://doi.org/10.1007/s10955-023-03073-y

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