# American Institute of Mathematical Sciences

August  2020, 40(8): 4665-4687. doi: 10.3934/dcds.2020197

## Fluctuations of ergodic sums on periodic orbits under specification

 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-909, Brazil 3 Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil

Received  April 2019 Published  May 2020

We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.

Citation: Manfred Denker, Samuel Senti, Xuan Zhang. Fluctuations of ergodic sums on periodic orbits under specification. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4665-4687. doi: 10.3934/dcds.2020197
##### References:
 [1] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114. [2] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.  doi: 10.1090/S0002-9947-1993-1107025-2. [3] R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452. [4] R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279. [6] R. C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, UT, 2007. [7] R. Burton and M. Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc., 302 (1987), 715-726.  doi: 10.1090/S0002-9947-1987-0891642-6. [8] M. Denker, The central limit theorem for dynamical systems, in Dynamical Systems and Ergodic Theory, Banach Center Publ., 23, PWN, Warsaw, 1989, 33–62. [9] M. Denker, J. Duan and M. McCourt, Pseudorandom numbers for conformal measures, Dyn. Syst., 24 (2009), 439-457.  doi: 10.1080/14689360903002019. [10] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364. [11] M. Denker and G. Keller, On $U$-statistics and v. Mises' statistics for weakly dependent processes, Z. Wahrsch. Verw. Gebiete, 64 (1983), 505-522.  doi: 10.1007/BF00534953. [12] M. Denker, S. Senti and X. Zhang, The Lindeberg theorem for Gibbs-Markov dynamics, Nonlinearity, 30 (2017), 4587-4613.  doi: 10.1088/1361-6544/aa8ca2. [13] P. Doukhan, Mixing. Properties and Examples, Lecture Notes in Statistics, 85, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2642-0. [14] M. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR, 188 (1969), 739-741. [15] S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields, 128 (2004), 82-122.  doi: 10.1007/s00440-003-0300-4. [16] S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electron. J. Probab., 19 (2014), 30pp. doi: 10.1214/EJP.v19-3427. [17] B. Hasselblatt, Introduction to hyperbolic dynamics and ergodic theory, in Ergodic Theory and Negative Curvature, Lecture Notes in Math, 2164, Springer, Cham, 2017, 1–124. doi: 10.1007/978-3-319-43059-1_1. [18] N. Haydn, M. Nicol, S. Vaienti and L. Zhang, Central limit theorems for the shrinking target problem, J. Stat. Phys., 153 (2013), 864-887.  doi: 10.1007/s10955-013-0860-3. [19] H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874. [20] I. A. Ibragimov and Y. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971,473pp. [21] A. Klenke, Probability Theory, Universitext, Springer, London, 2014. doi: 10.1007/978-1-4471-5361-0. [22] D. Kwietniak, M. Łącka and P. Oprocha, A panorama of specification-like properties and their consequences, in Dynamics and Numbers, Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016,155–186. doi: 10.1090/conm/669/13428. [23] C. Liverani, Central limit theorem for deterministic systems, in International Conference on Dynamical Systems), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996, 56–75. [24] R. Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems, 5 (1985), 71-88.  doi: 10.1017/S0143385700002765. [25] V. V. Petrov, Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer-Verlag, New York-Heidelberg, 1975. doi: 10.1007/978-3-642-65809-9. [26] J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab., 11 (1983), 772-788.  doi: 10.1214/aop/1176993522. [27] D. Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity, 5 (1992), 1223-1236.  doi: 10.1088/0951-7715/5/6/002. [28] K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606. [29] D. Thomine, A generalized central limit theorem in infinite ergodic theory, Probab. Theory Related Fields, 158 (2014), 597-636.  doi: 10.1007/s00440-013-0491-2. [30] D. Thomine, Variations on a central limit theorem in infinite ergodic theory, Ergodic Theory Dynam. Systems, 35 (2015), 1610-1657.  doi: 10.1017/etds.2013.114.

show all references

##### References:
 [1] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114. [2] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.  doi: 10.1090/S0002-9947-1993-1107025-2. [3] R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452. [4] R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279. [6] R. C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, UT, 2007. [7] R. Burton and M. Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc., 302 (1987), 715-726.  doi: 10.1090/S0002-9947-1987-0891642-6. [8] M. Denker, The central limit theorem for dynamical systems, in Dynamical Systems and Ergodic Theory, Banach Center Publ., 23, PWN, Warsaw, 1989, 33–62. [9] M. Denker, J. Duan and M. McCourt, Pseudorandom numbers for conformal measures, Dyn. Syst., 24 (2009), 439-457.  doi: 10.1080/14689360903002019. [10] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364. [11] M. Denker and G. Keller, On $U$-statistics and v. Mises' statistics for weakly dependent processes, Z. Wahrsch. Verw. Gebiete, 64 (1983), 505-522.  doi: 10.1007/BF00534953. [12] M. Denker, S. Senti and X. Zhang, The Lindeberg theorem for Gibbs-Markov dynamics, Nonlinearity, 30 (2017), 4587-4613.  doi: 10.1088/1361-6544/aa8ca2. [13] P. Doukhan, Mixing. Properties and Examples, Lecture Notes in Statistics, 85, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2642-0. [14] M. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR, 188 (1969), 739-741. [15] S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields, 128 (2004), 82-122.  doi: 10.1007/s00440-003-0300-4. [16] S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electron. J. Probab., 19 (2014), 30pp. doi: 10.1214/EJP.v19-3427. [17] B. Hasselblatt, Introduction to hyperbolic dynamics and ergodic theory, in Ergodic Theory and Negative Curvature, Lecture Notes in Math, 2164, Springer, Cham, 2017, 1–124. doi: 10.1007/978-3-319-43059-1_1. [18] N. Haydn, M. Nicol, S. Vaienti and L. Zhang, Central limit theorems for the shrinking target problem, J. Stat. Phys., 153 (2013), 864-887.  doi: 10.1007/s10955-013-0860-3. [19] H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874. [20] I. A. Ibragimov and Y. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971,473pp. [21] A. Klenke, Probability Theory, Universitext, Springer, London, 2014. doi: 10.1007/978-1-4471-5361-0. [22] D. Kwietniak, M. Łącka and P. Oprocha, A panorama of specification-like properties and their consequences, in Dynamics and Numbers, Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016,155–186. doi: 10.1090/conm/669/13428. [23] C. Liverani, Central limit theorem for deterministic systems, in International Conference on Dynamical Systems), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996, 56–75. [24] R. Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems, 5 (1985), 71-88.  doi: 10.1017/S0143385700002765. [25] V. V. Petrov, Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer-Verlag, New York-Heidelberg, 1975. doi: 10.1007/978-3-642-65809-9. [26] J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab., 11 (1983), 772-788.  doi: 10.1214/aop/1176993522. [27] D. Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity, 5 (1992), 1223-1236.  doi: 10.1088/0951-7715/5/6/002. [28] K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606. [29] D. Thomine, A generalized central limit theorem in infinite ergodic theory, Probab. Theory Related Fields, 158 (2014), 597-636.  doi: 10.1007/s00440-013-0491-2. [30] D. Thomine, Variations on a central limit theorem in infinite ergodic theory, Ergodic Theory Dynam. Systems, 35 (2015), 1610-1657.  doi: 10.1017/etds.2013.114.
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