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Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions
$ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization
1. | LAMFA, UMR 7352 CNRS, University of Picardie, 33 rue Saint Leu, 80039 Amiens, France |
2. | School of Mathematics and Statistics, Central China Normal University 430079 Wuhan, China |
3. | Centre for Mathematical Sciences, Lund University, Box 118, 221 00 LUND, Sweden |
4. | Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, 200438 Shanghai, China |
Given an integer $ q\ge 2 $ and a real number $ c\in [0,1) $, consider the generalized Thue-Morse sequence $ (t_n^{(q;c)})_{n\ge 0} $ defined by $ t_n^{(q;c)} = e^{2\pi i c s_q(n)} $, where $ s_q(n) $ is the sum of digits of the $ q $-expansion of $ n $. We prove that the $ L^\infty $-norm of the trigonometric polynomials $ \sigma_{N}^{(q;c)} (x) : = \sum_{n = 0}^{N-1} t_n^{(q;c)} e^{2\pi i n x} $, behaves like $ N^{\gamma(q;c)} $, where $ \gamma(q;c) $ is equal to the dynamical maximal value of $ \log_q \left|\frac{\sin q\pi (x+c)}{\sin \pi (x+c)}\right| $ relative to the dynamics $ x \mapsto qx \mod 1 $ and that the maximum value is attained by a $ q $-Sturmian measure. Numerical values of $ \gamma(q;c) $ can be computed.
References:
[1] |
C. Aistleitner, R. Hofer and G. Larcher, On evil Kronecker sequences and lacunary trigonometric products, Ann. Inst. Fourier (Grenoble), 67 (2017), 637–687.
doi: 10.5802/aif.3094. |
[2] |
V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard, Entrance time functions for flat spot maps, Nonlinearity, 23 (2010), 1477–1494.
doi: 10.1088/0951-7715/23/6/011. |
[3] |
V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, The flat spot standard family: Variation of the entrance time median, Dyn. Syst., 27 (2012), 29–43.
doi: 10.1080/14689367.2011.625553. |
[4] |
V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, Sturmian maximizing measures for the piecewise-linear cosine family, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 285–302.
doi: 10.1007/s00574-012-0013-3. |
[5] |
J. Bochi, Ergodic opitimization of Birkhoff averages and Lyapunov exponents, Proc. Int. Cong. Math. 2018 Rio de Janeiro, 3 (2018), 1825–1846. |
[6] |
T. Bousch, Le poisson n'a pas d'arêtes, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 489–508.
doi: 10.1016/S0246-0203(00)00132-1. |
[7] |
T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287–311.
doi: 10.1016/S0012-9593(00)01062-4. |
[8] |
T. Bousch and O. Jenkinson, Cohomology classes of dynamically non-negative $C^k$ functions, Invent. Math., 148 (2002), 207–217.
doi: 10.1007/s002220100194. |
[9] |
C. Boyd, On the structure of the family of Cherry fields on the torus, Ergodic Theory Dynam. Systems, 5 (1985), 27–46.
doi: 10.1017/S014338570000273X. |
[10] |
S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc., 115 (1994), 451–481.
doi: 10.1017/S0305004100072236. |
[11] |
G. Contreras, A. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1379–1409.
doi: 10.1017/S0143385701001663. |
[12] |
G. Contreras, Ground states are generically a periodic orbit, Invent. Math., 205 (2016), 383–412.
doi: 10.1007/s00222-015-0638-0. |
[13] |
J. P. Conze and Yves Guivarc'h, Croissance des sommes ergodiques et principe variationnel, Unpublished preprint. Google Scholar |
[14] |
C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d'entiers, Ann. Inst. Fourier (Grenoble), 55 (2005), 2423–2474.
doi: 10.5802/aif.2166. |
[15] |
A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory Dynam. Systems, 39 (2019), 1275–1289.
doi: 10.1017/etds.2017.81. |
[16] |
A. Fan, J. Schmeling and W. Shen, Multifractal analysis of generalized Thue-Morse polynomials, In preparation. Google Scholar |
[17] |
A. Fan and J. Konieczny, On uniformity of $q$-multiplicative sequences, Bull. Lond. Math. Soc., 51 (2019), 466–488.
doi: 10.1112/blms.12245. |
[18] |
E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres, Acta Arith., 77 (1996), 339–351.
doi: 10.4064/aa-77-4-339-351. |
[19] |
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, Math. Ann., 305 (1996), 571–599.
doi: 10.1007/BF01444238. |
[20] |
A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259–265.
doi: 10.4064/aa-13-3-259-265. |
[21] |
M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233. |
[22] |
O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems, 39 (2019), 2593–2618.
doi: 10.1017/etds.2017.142. |
[23] |
O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197–224.
doi: 10.3934/dcds.2006.15.197. |
[24] |
O. Jenkinson, Optimization and majorization of invariant measures, Electron. Res. Announc. Amer. Math. Soc., 13 (2007), 1–12.
doi: 10.1090/S1079-6762-07-00170-9. |
[25] |
O. Jenkinson, A partial order on $\times2$-invariant measures, Math. Res. Lett., 15 (2008), 893–900.
doi: 10.4310/MRL.2008.v15.n5.a6. |
[26] |
O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl., 1 (2009), 463–483.
doi: 10.1142/S179383090900035X. |
[27] |
O. Jenkinson, R. D. Mauldin and M. Urbański, Ergodic optimization for noncompact dynamical systems, Dyn. Syst., 22 (2007), 379–388.
doi: 10.1080/14689360701450543. |
[28] |
O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures and the finiteness conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 3062–3100.
doi: 10.1017/etds.2017.18. |
[29] |
O. Jenkinson and J. Steel, Majorization of invariant measures for orientation-reversing maps, Ergodic Theory Dynam. Systems, 30 (2010), 1471–1483.
doi: 10.1017/S0143385709000686. |
[30] |
J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, Ann. Inst. Fourier (Grenoble), 69 (2019), 1897–1913.
doi: 10.5802/aif.3285. |
[31] |
K. Mahler, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions: Part two on the translation properties of a simple class of arithmetical functions, Journal of Mathematics and Physics, 6 (1927), 158–163.
doi: 10.1002/sapm192761158. |
[32] |
C. Mauduit and J. Rivat, La somme des chiffres des carrés, Acta Math., 203 (2009), 107–148.
doi: 10.1007/s11511-009-0040-0. |
[33] |
C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. (2), 171 (2010), 1591–1646.
doi: 10.4007/annals.2010.171.1591. |
[34] |
C. Mauduit, J. Rivat and A. Sárközy, On the digits of sumsets, Canad. J. Math., 69 (2017), 595–612.
doi: 10.4153/CJM-2016-007-2. |
[35] |
V. A. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268–282. |
[36] |
M. Queffélec, Questions around the Thue-Morse sequence, Unif. Distrib. Theory, 13 (2018), 1–25.
doi: 10.1515/udt-2018-0001. |
[37] |
J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419–428.
doi: 10.1088/0951-7715/2/3/003. |
[38] |
Y. Zhang, K. Yin and W. Wu, A rigorous computer aided estimation for Gelfond exponent of weighted Thue-Morse sequences, arXiv: 1806.08329v2. Google Scholar |
show all references
References:
[1] |
C. Aistleitner, R. Hofer and G. Larcher, On evil Kronecker sequences and lacunary trigonometric products, Ann. Inst. Fourier (Grenoble), 67 (2017), 637–687.
doi: 10.5802/aif.3094. |
[2] |
V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard, Entrance time functions for flat spot maps, Nonlinearity, 23 (2010), 1477–1494.
doi: 10.1088/0951-7715/23/6/011. |
[3] |
V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, The flat spot standard family: Variation of the entrance time median, Dyn. Syst., 27 (2012), 29–43.
doi: 10.1080/14689367.2011.625553. |
[4] |
V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, Sturmian maximizing measures for the piecewise-linear cosine family, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 285–302.
doi: 10.1007/s00574-012-0013-3. |
[5] |
J. Bochi, Ergodic opitimization of Birkhoff averages and Lyapunov exponents, Proc. Int. Cong. Math. 2018 Rio de Janeiro, 3 (2018), 1825–1846. |
[6] |
T. Bousch, Le poisson n'a pas d'arêtes, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 489–508.
doi: 10.1016/S0246-0203(00)00132-1. |
[7] |
T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287–311.
doi: 10.1016/S0012-9593(00)01062-4. |
[8] |
T. Bousch and O. Jenkinson, Cohomology classes of dynamically non-negative $C^k$ functions, Invent. Math., 148 (2002), 207–217.
doi: 10.1007/s002220100194. |
[9] |
C. Boyd, On the structure of the family of Cherry fields on the torus, Ergodic Theory Dynam. Systems, 5 (1985), 27–46.
doi: 10.1017/S014338570000273X. |
[10] |
S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc., 115 (1994), 451–481.
doi: 10.1017/S0305004100072236. |
[11] |
G. Contreras, A. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1379–1409.
doi: 10.1017/S0143385701001663. |
[12] |
G. Contreras, Ground states are generically a periodic orbit, Invent. Math., 205 (2016), 383–412.
doi: 10.1007/s00222-015-0638-0. |
[13] |
J. P. Conze and Yves Guivarc'h, Croissance des sommes ergodiques et principe variationnel, Unpublished preprint. Google Scholar |
[14] |
C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d'entiers, Ann. Inst. Fourier (Grenoble), 55 (2005), 2423–2474.
doi: 10.5802/aif.2166. |
[15] |
A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory Dynam. Systems, 39 (2019), 1275–1289.
doi: 10.1017/etds.2017.81. |
[16] |
A. Fan, J. Schmeling and W. Shen, Multifractal analysis of generalized Thue-Morse polynomials, In preparation. Google Scholar |
[17] |
A. Fan and J. Konieczny, On uniformity of $q$-multiplicative sequences, Bull. Lond. Math. Soc., 51 (2019), 466–488.
doi: 10.1112/blms.12245. |
[18] |
E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres, Acta Arith., 77 (1996), 339–351.
doi: 10.4064/aa-77-4-339-351. |
[19] |
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, Math. Ann., 305 (1996), 571–599.
doi: 10.1007/BF01444238. |
[20] |
A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259–265.
doi: 10.4064/aa-13-3-259-265. |
[21] |
M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233. |
[22] |
O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems, 39 (2019), 2593–2618.
doi: 10.1017/etds.2017.142. |
[23] |
O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197–224.
doi: 10.3934/dcds.2006.15.197. |
[24] |
O. Jenkinson, Optimization and majorization of invariant measures, Electron. Res. Announc. Amer. Math. Soc., 13 (2007), 1–12.
doi: 10.1090/S1079-6762-07-00170-9. |
[25] |
O. Jenkinson, A partial order on $\times2$-invariant measures, Math. Res. Lett., 15 (2008), 893–900.
doi: 10.4310/MRL.2008.v15.n5.a6. |
[26] |
O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl., 1 (2009), 463–483.
doi: 10.1142/S179383090900035X. |
[27] |
O. Jenkinson, R. D. Mauldin and M. Urbański, Ergodic optimization for noncompact dynamical systems, Dyn. Syst., 22 (2007), 379–388.
doi: 10.1080/14689360701450543. |
[28] |
O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures and the finiteness conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 3062–3100.
doi: 10.1017/etds.2017.18. |
[29] |
O. Jenkinson and J. Steel, Majorization of invariant measures for orientation-reversing maps, Ergodic Theory Dynam. Systems, 30 (2010), 1471–1483.
doi: 10.1017/S0143385709000686. |
[30] |
J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, Ann. Inst. Fourier (Grenoble), 69 (2019), 1897–1913.
doi: 10.5802/aif.3285. |
[31] |
K. Mahler, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions: Part two on the translation properties of a simple class of arithmetical functions, Journal of Mathematics and Physics, 6 (1927), 158–163.
doi: 10.1002/sapm192761158. |
[32] |
C. Mauduit and J. Rivat, La somme des chiffres des carrés, Acta Math., 203 (2009), 107–148.
doi: 10.1007/s11511-009-0040-0. |
[33] |
C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. (2), 171 (2010), 1591–1646.
doi: 10.4007/annals.2010.171.1591. |
[34] |
C. Mauduit, J. Rivat and A. Sárközy, On the digits of sumsets, Canad. J. Math., 69 (2017), 595–612.
doi: 10.4153/CJM-2016-007-2. |
[35] |
V. A. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268–282. |
[36] |
M. Queffélec, Questions around the Thue-Morse sequence, Unif. Distrib. Theory, 13 (2018), 1–25.
doi: 10.1515/udt-2018-0001. |
[37] |
J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419–428.
doi: 10.1088/0951-7715/2/3/003. |
[38] |
Y. Zhang, K. Yin and W. Wu, A rigorous computer aided estimation for Gelfond exponent of weighted Thue-Morse sequences, arXiv: 1806.08329v2. Google Scholar |



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