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June  2022, 42(6): 2699-2718. doi: 10.3934/dcds.2021208

On involution kernels and large deviations principles on $\beta$-shifts

 School of Mathematics, National University of Colombia, Medellín 050034, Colombia

Received  January 2021 Revised  October 2021 Published  June 2022 Early access  January 2022

Fund Project: Supported by FFJC-MINCIENCIAS Process 80740-628-2020

Consider $\beta > 1$ and $\lfloor \beta \rfloor$ its integer part. It is widely known that any real number $\alpha \in \Bigl[0, \frac{\lfloor \beta \rfloor}{\beta - 1}\Bigr]$ can be represented in base $\beta$ using a development in series of the form $\alpha = \sum_{n = 1}^\infty x_n\beta^{-n}$, where $x = (x_n)_{n \geq 1}$ is a sequence taking values into the alphabet $\{0,\; ...\; ,\; \lfloor \beta \rfloor\}$. The so called $\beta$-shift, denoted by $\Sigma_\beta$, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy $\beta$-expansion of $1$. Fixing a Hölder continuous potential $A$, we show an explicit expression for the main eigenfunction of the Ruelle operator $\psi_A$, in order to obtain a natural extension to the bilateral $\beta$-shift of its corresponding Gibbs state $\mu_A$. Our main goal here is to prove a first level large deviations principle for the family $(\mu_{tA})_{t>1}$ with a rate function $I$ attaining its maximum value on the union of the supports of all the maximizing measures of $A$. The above is proved through a technique using the representation of $\Sigma_\beta$ and its bilateral extension $\widehat{\Sigma_\beta}$ in terms of the quasi-greedy $\beta$-expansion of $1$ and the so called involution kernel associated to the potential $A$.

Citation: Victor Vargas. On involution kernels and large deviations principles on $\beta$-shifts. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2699-2718. doi: 10.3934/dcds.2021208
References:
 [1] A. Baraviera, R. Leplaideur and A. Lopes, Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013. 29o Colóquio Brasileiro de Matemática. [29th Brazilian Mathematics Colloquium]. [2] A. Baraviera, A. O. Lopes and P. Thieullen, A large deviation principle for the equilibrium states of Hölder potentials: The zero temperature case, Stoch. Dyn., 6 (2006), 77-96.  doi: 10.1142/S0219493706001657. [3] A. T. Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza, On the general one-dimensional $XY$ model: Positive and zero temperature, selection and non-selection, Rev. Math. Phys., 23 (2011), 1063-1113.  doi: 10.1142/S0129055X11004527. [4] A. Bertrand-Mathis, Développement en base $\theta$; répartition modulo un de la suite $(x\theta^n)_{n\geq 0}$; langages codés et $\theta$-shift, Bull. Soc. Math. France, 114 (1986), 271-323.  doi: 10.24033/bsmf.2058. [5] R. Bissacot, J. K. Mengue and E. Pérez, A large deviation principle for gibbs states on countable markov shifts at zero temperature, 2015. arXiv: 1612.05831. [6] F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  doi: 10.1016/0304-3975(89)90038-8. [7] V. Climenhaga, D. J. Thompson and K. Yamamoto, Large deviations for systems with non-uniform structure, Trans. Amer. Math. Soc., 369 (2017), 4167-4192.  doi: 10.1090/tran/6786. [8] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, volume 38 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. doi: 10.1007/978-3-642-03311-7. [9] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, volume 527, Springer, Cham, 1976. doi: 10.1007/BFb0082364. [10] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. Ⅰ. Ⅱ, Comm. Pure Appl. Math., 28, 1–47; ibid. 28 (1975), 279–301. doi: 10.1002/cpa.3160280102. [11] R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, , Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1985 original. doi: 10.1007/3-540-29060-5. [12] P. Erdős, M. Horváth and I. Joó, On the uniqueness of the expansions $1 = \sum q^{-n_i}$, Acta Math. Hungar., 58 (1991), 333-342.  doi: 10.1007/BF01903963. [13] P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions $1 = \sum^\infty_{i = 1}q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.  doi: 10.24033/bsmf.2151. [14] A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators. Ⅰ. Ruelle theorem, Comm. Math. Phys., 223 (2001), 125-141.  doi: 10.1007/s002200100538. [15] A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators. Ⅱ. Convergence speeds, Comm. Math. Phys., 223 (2001), 143-159.  doi: 10.1007/s002200100539. [16] P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543.  doi: 10.4310/MRL.2001.v8.n4.a12. [17] B. P. Kitchens, Symbolic Dynamics, , Universitext. Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. doi: 10.1007/978-3-642-58822-8. [18] A. O. Lopes and J. K. Mengue, Selection of measure and a large deviation principle for the general one-dimensional $XY$ model, Dyn. Syst., 29 (2014), 24-39.  doi: 10.1080/14689367.2013.835792. [19] A. O. Lopes, J. K. Mengue, J. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: Positive and zero temperature, Ergodic Theory Dynam. Systems, 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15. [20] A. O. Lopes and V. Vargas, Gibbs states and Gibbsian specifications on the space $\mathbb{R}^\mathbb{N}$, Dyn. Syst., 35 (2020), 216–241. doi: 10.1080/14689367.2019.1663789. [21] W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954. [22] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 268 (1990), 187-188. [23] C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta$-shifts, Nonlinearity, 18 (2005), 237-261.  doi: 10.1088/0951-7715/18/1/013. [24] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331. [25] N. Sidorov, Almost every number has a continuum of $\beta$-expansions, Am. Math. Mon., 110 (2003), 838–842. doi: 10.2307/3647804. [26] R. R. Souza and V. Vargas, Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure, Qual. Theory Dyn. Syst., 21 (2022), Paper No. 5, 28 pp. doi: 10.1007/s12346-021-00537-y. [27] S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286.  doi: 10.1002/cpa.3160190303. [28] C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058. [29] P. Walters, Equilibrium states for $\beta$-transformations and related transformations, Math. Z., 159 (1978), 65-88.  doi: 10.1007/BF01174569. [30] P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

show all references

References:
 [1] A. Baraviera, R. Leplaideur and A. Lopes, Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013. 29o Colóquio Brasileiro de Matemática. [29th Brazilian Mathematics Colloquium]. [2] A. Baraviera, A. O. Lopes and P. Thieullen, A large deviation principle for the equilibrium states of Hölder potentials: The zero temperature case, Stoch. Dyn., 6 (2006), 77-96.  doi: 10.1142/S0219493706001657. [3] A. T. Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza, On the general one-dimensional $XY$ model: Positive and zero temperature, selection and non-selection, Rev. Math. Phys., 23 (2011), 1063-1113.  doi: 10.1142/S0129055X11004527. [4] A. Bertrand-Mathis, Développement en base $\theta$; répartition modulo un de la suite $(x\theta^n)_{n\geq 0}$; langages codés et $\theta$-shift, Bull. Soc. Math. France, 114 (1986), 271-323.  doi: 10.24033/bsmf.2058. [5] R. Bissacot, J. K. Mengue and E. Pérez, A large deviation principle for gibbs states on countable markov shifts at zero temperature, 2015. arXiv: 1612.05831. [6] F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  doi: 10.1016/0304-3975(89)90038-8. [7] V. Climenhaga, D. J. Thompson and K. Yamamoto, Large deviations for systems with non-uniform structure, Trans. Amer. Math. Soc., 369 (2017), 4167-4192.  doi: 10.1090/tran/6786. [8] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, volume 38 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. doi: 10.1007/978-3-642-03311-7. [9] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, volume 527, Springer, Cham, 1976. doi: 10.1007/BFb0082364. [10] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. Ⅰ. Ⅱ, Comm. Pure Appl. Math., 28, 1–47; ibid. 28 (1975), 279–301. doi: 10.1002/cpa.3160280102. [11] R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, , Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1985 original. doi: 10.1007/3-540-29060-5. [12] P. Erdős, M. Horváth and I. Joó, On the uniqueness of the expansions $1 = \sum q^{-n_i}$, Acta Math. Hungar., 58 (1991), 333-342.  doi: 10.1007/BF01903963. [13] P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions $1 = \sum^\infty_{i = 1}q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.  doi: 10.24033/bsmf.2151. [14] A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators. Ⅰ. Ruelle theorem, Comm. Math. Phys., 223 (2001), 125-141.  doi: 10.1007/s002200100538. [15] A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators. Ⅱ. Convergence speeds, Comm. Math. Phys., 223 (2001), 143-159.  doi: 10.1007/s002200100539. [16] P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543.  doi: 10.4310/MRL.2001.v8.n4.a12. [17] B. P. Kitchens, Symbolic Dynamics, , Universitext. Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. doi: 10.1007/978-3-642-58822-8. [18] A. O. Lopes and J. K. Mengue, Selection of measure and a large deviation principle for the general one-dimensional $XY$ model, Dyn. Syst., 29 (2014), 24-39.  doi: 10.1080/14689367.2013.835792. [19] A. O. Lopes, J. K. Mengue, J. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: Positive and zero temperature, Ergodic Theory Dynam. Systems, 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15. [20] A. O. Lopes and V. Vargas, Gibbs states and Gibbsian specifications on the space $\mathbb{R}^\mathbb{N}$, Dyn. Syst., 35 (2020), 216–241. doi: 10.1080/14689367.2019.1663789. [21] W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954. [22] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 268 (1990), 187-188. [23] C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta$-shifts, Nonlinearity, 18 (2005), 237-261.  doi: 10.1088/0951-7715/18/1/013. [24] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331. [25] N. Sidorov, Almost every number has a continuum of $\beta$-expansions, Am. Math. Mon., 110 (2003), 838–842. doi: 10.2307/3647804. [26] R. R. Souza and V. Vargas, Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure, Qual. Theory Dyn. Syst., 21 (2022), Paper No. 5, 28 pp. doi: 10.1007/s12346-021-00537-y. [27] S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286.  doi: 10.1002/cpa.3160190303. [28] C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058. [29] P. Walters, Equilibrium states for $\beta$-transformations and related transformations, Math. Z., 159 (1978), 65-88.  doi: 10.1007/BF01174569. [30] P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
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