doi: 10.3934/dcds.2021208
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems, 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].  Google Scholar

[2]

A. BaravieraA. 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.  Google Scholar

[3]

A. T. BaravieraL. M. CiolettiA. O. LopesJ. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[6]

F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  doi: 10.1016/0304-3975(89)90038-8.  Google Scholar

[7]

V. ClimenhagaD. 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.  Google Scholar

[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.  Google Scholar

[9]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, volume 527, Springer, Cham, 1976. doi: 10.1007/BFb0082364.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. ErdősM. 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.  Google Scholar

[13]

P. ErdösI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. O. LopesJ. K. MengueJ. 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.  Google Scholar

[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.  Google Scholar

[21]

W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[22]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 268 (1990), 187-188.   Google Scholar

[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.  Google Scholar

[24]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

[25]

N. Sidorov, Almost every number has a continuum of $\beta$-expansions, Am. Math. Mon., 110 (2003), 838–842. doi: 10.2307/3647804.  Google Scholar

[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.  Google Scholar

[27]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286.  doi: 10.1002/cpa.3160190303.  Google Scholar

[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.  Google Scholar

[29]

P. Walters, Equilibrium states for $\beta $-transformations and related transformations, Math. Z., 159 (1978), 65-88.  doi: 10.1007/BF01174569.  Google Scholar

[30]

P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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].  Google Scholar

[2]

A. BaravieraA. 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.  Google Scholar

[3]

A. T. BaravieraL. M. CiolettiA. O. LopesJ. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[6]

F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  doi: 10.1016/0304-3975(89)90038-8.  Google Scholar

[7]

V. ClimenhagaD. 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.  Google Scholar

[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.  Google Scholar

[9]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, volume 527, Springer, Cham, 1976. doi: 10.1007/BFb0082364.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. ErdősM. 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.  Google Scholar

[13]

P. ErdösI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. O. LopesJ. K. MengueJ. 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.  Google Scholar

[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.  Google Scholar

[21]

W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[22]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 268 (1990), 187-188.   Google Scholar

[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.  Google Scholar

[24]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

[25]

N. Sidorov, Almost every number has a continuum of $\beta$-expansions, Am. Math. Mon., 110 (2003), 838–842. doi: 10.2307/3647804.  Google Scholar

[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.  Google Scholar

[27]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286.  doi: 10.1002/cpa.3160190303.  Google Scholar

[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.  Google Scholar

[29]

P. Walters, Equilibrium states for $\beta $-transformations and related transformations, Math. Z., 159 (1978), 65-88.  doi: 10.1007/BF01174569.  Google Scholar

[30]

P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[1]

Leandro Cioletti, Artur O. Lopes, Manuel Stadlbauer. Ruelle operator for continuous potentials and DLR-Gibbs measures. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4625-4652. doi: 10.3934/dcds.2020195

[2]

Vesselin Petkov, Luchezar Stoyanov. Spectral estimates for Ruelle operators with two parameters and sharp large deviations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6391-6417. doi: 10.3934/dcds.2019277

[3]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523

[4]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729

[5]

Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773

[6]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113

[7]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881

[8]

Eugen Mihailescu. Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 961-975. doi: 10.3934/dcds.2012.32.961

[9]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103

[10]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805

[11]

Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021021

[12]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216

[13]

Leandro Cioletti, Artur O. Lopes. Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6139-6152. doi: 10.3934/dcds.2017264

[14]

Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016

[15]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327

[16]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155

[17]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245

[18]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965

[19]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221

[20]

Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (37)
  • HTML views (28)
  • Cited by (0)

Other articles
by authors

[Back to Top]