January  2021, 41(1): 297-327. doi: 10.3934/dcds.2020363

$ 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

Received  October 2019 Revised  August 2020 Published  January 2021 Early access  October 2020

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.

Citation: Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363
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.

[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.

[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.

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.

[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.

[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.

Figure 1.  The graphs of $ f_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $
Figure 2.  The graphs of $ \gamma(2;c) $. Only the cycles of order $ \le 13 $ are used to plot the graph. To fill in the gaps in the graph, we have to use other cycles (there are infinitely many)
Figure 3.  The graphs of $ \log|2\sin \pi (x-b)| $ on the intervals $ [0,1] $ and $ [b,b+1] $ with $ b = 1/3 $
Figure 4.  The graphs of $ f'_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $
Figure 5.  The branch $ T|_{C_\lambda} $
Figure 6.  The graphs of $ e_0 $ and $ e_{1/4} $
Table 1.  Values of $ \beta(c) $ and $ \gamma(c) $ for specific $ c $'s
$c$ $\beta(c)$ $ \gamma(c)$ $ c $ $ \beta(c)$ $\gamma(c)$
$1/2$ $\log(\sqrt{3})$ $ \log 3/\log 4$ $7/18$ $0.51079$ $ 0.73691 $
$1/3$ $0.52227$ $0.75347$ $4/19$ $0.51949$ $0.74947$
$1/4$ $0.51586$ $0.74423$ $5/19$ $0.51719$ $0.74615$
$1/5$ $0.52201$ $0.75310$ $6/19$ $0.51830$ $0.74775$
$2/5$ $0.51217$ $0.73890$ $7/19$ $0.51701$ $0.74589$
$2/7$ $0.51354$ $0.74088$ $8/19$ $0.51252$ $0.73941$
$3/7$ $0.51515$ $0.74321$ $9/19$ $0.54474$ $0.78589$
$3/8$ $0.51406$ $0.74163$ $7/20$ $0.52195$ $0.75302$
$2/9$ $0.51848$ $0.74802$ $9/20$ $0.53272$ $0.76855$
$4/9$ $0.52879$ $0.76288$ $4/21$ $0.52489$ $0.75725$
$3/10$ $0.51184$ $0.73843$ $5/21$ $0.51576$ $0.74408$
$2/11$ $0.52852$ $0.76250$ $8/21$ **
$3/11$ $0.51655$ $0.74523$ $5/22$ $0.51802$ $0.74735$
$4/11$ $0.51875$ $0.74840$ $7/22$ $0.51910$ $0.74891$
$5/11$ $0.53562$ $0.77273$ $9/22$ $0.51196$ $0.73860$
$5/12$ $0.51185$ $0.73844$ $5/23$ $0.51857$ $0.74814$
$3/13$ $0.51748$ $0.74657$ $6/23$ $0.51714$ $0.74608$
$4/13$ $0.51496$ $0.74293$ $7/23$ $0.51329$ $0.74052$
$5/13$ $0.49827$ $0.71885$ $8/23$ $0.52222$ $0.75340$
$6/13$ $0.53952$ $0.77837$ $9/23$ $0.51124$ $0.73756$
$3/14$ $0.51844$ $0.74795$ $10/23$ $0.52092$ $0.75153$
$5/14$ $0.52061$ $0.75108$ $11/23$ $0.54619$ $0.78799$
$7/15$ $0.54197$ $0.78190$ $5/24$ $0.52015$ $0.75042$
$7/16$ $0.52326$ $0.75491$ $7/24$ $0.51179$ $0.73836$
$3/17$ $0.53203$ $0.76756$ $11/24$ $0.53782$ $0.77591$
$4/17$ $0.51651$ $0.74516$ $6/25$ $0.51517$ $0.74324$
$5/17$ $0.51191$ $0.73853$ $7/25$ $0.515168$ $0.74323$
$6/17$ $0.52148$ $0.75234$ $8/25$ $0.51966$ $0.74971$
$7/17$ $0.51167$ $0.73818$ $9/25$ $0.51987$ $0.75001$
$8/17$ $0.54360$ $0.78425$ $11/25$ $0.52534$ $0.75789$
$5/18$ $0.51567$ $0.74396$ $12/25$ $0.54667$ $0.78868$
  ** We don't compute $\beta(c)$ and $\gamma(c)$ if the parameter $c$ doesn't belong to any of the intervals in Table 2.
$c$ $\beta(c)$ $ \gamma(c)$ $ c $ $ \beta(c)$ $\gamma(c)$
$1/2$ $\log(\sqrt{3})$ $ \log 3/\log 4$ $7/18$ $0.51079$ $ 0.73691 $
$1/3$ $0.52227$ $0.75347$ $4/19$ $0.51949$ $0.74947$
$1/4$ $0.51586$ $0.74423$ $5/19$ $0.51719$ $0.74615$
$1/5$ $0.52201$ $0.75310$ $6/19$ $0.51830$ $0.74775$
$2/5$ $0.51217$ $0.73890$ $7/19$ $0.51701$ $0.74589$
$2/7$ $0.51354$ $0.74088$ $8/19$ $0.51252$ $0.73941$
$3/7$ $0.51515$ $0.74321$ $9/19$ $0.54474$ $0.78589$
$3/8$ $0.51406$ $0.74163$ $7/20$ $0.52195$ $0.75302$
$2/9$ $0.51848$ $0.74802$ $9/20$ $0.53272$ $0.76855$
$4/9$ $0.52879$ $0.76288$ $4/21$ $0.52489$ $0.75725$
$3/10$ $0.51184$ $0.73843$ $5/21$ $0.51576$ $0.74408$
$2/11$ $0.52852$ $0.76250$ $8/21$ **
$3/11$ $0.51655$ $0.74523$ $5/22$ $0.51802$ $0.74735$
$4/11$ $0.51875$ $0.74840$ $7/22$ $0.51910$ $0.74891$
$5/11$ $0.53562$ $0.77273$ $9/22$ $0.51196$ $0.73860$
$5/12$ $0.51185$ $0.73844$ $5/23$ $0.51857$ $0.74814$
$3/13$ $0.51748$ $0.74657$ $6/23$ $0.51714$ $0.74608$
$4/13$ $0.51496$ $0.74293$ $7/23$ $0.51329$ $0.74052$
$5/13$ $0.49827$ $0.71885$ $8/23$ $0.52222$ $0.75340$
$6/13$ $0.53952$ $0.77837$ $9/23$ $0.51124$ $0.73756$
$3/14$ $0.51844$ $0.74795$ $10/23$ $0.52092$ $0.75153$
$5/14$ $0.52061$ $0.75108$ $11/23$ $0.54619$ $0.78799$
$7/15$ $0.54197$ $0.78190$ $5/24$ $0.52015$ $0.75042$
$7/16$ $0.52326$ $0.75491$ $7/24$ $0.51179$ $0.73836$
$3/17$ $0.53203$ $0.76756$ $11/24$ $0.53782$ $0.77591$
$4/17$ $0.51651$ $0.74516$ $6/25$ $0.51517$ $0.74324$
$5/17$ $0.51191$ $0.73853$ $7/25$ $0.515168$ $0.74323$
$6/17$ $0.52148$ $0.75234$ $8/25$ $0.51966$ $0.74971$
$7/17$ $0.51167$ $0.73818$ $9/25$ $0.51987$ $0.75001$
$8/17$ $0.54360$ $0.78425$ $11/25$ $0.52534$ $0.75789$
$5/18$ $0.51567$ $0.74396$ $12/25$ $0.54667$ $0.78868$
  ** We don't compute $\beta(c)$ and $\gamma(c)$ if the parameter $c$ doesn't belong to any of the intervals in Table 2.
Table 2.  Valid intervals $[c_*, c^*]$
Period $s_{\max}-\frac{1}{2}$ $s_{\min}$ $ [c_*,c^*] $
$1$ $-1/2$ $0$ $[0.000000000000000,0.175160000000000]$
$2$ $1/6$ $1/3$ $[0.428133329021334,0.571866670978666]$
$ 3 $ $1/14 $ $ 1/7 $ $ [0.619203577131485,0.697872156658965] $
$ 3 $ $5/14 $ $ 3/7 $ $ [0.302127843341035,0.380796422868515] $
$ 4 $ $1/30 $ $ 1/15 $ $ [0.709633870795466,0.755421357085333] $
$ 4 $ $13/30 $ $ 7/15 $ $ [0.244578642914667,0.290366129204534] $
$ 5 $ $1/62 $ $ 1/31 $ $ [0.758710839860046,0.785842721390351] $
$ 5 $ $29/62$ $15/31 $ $ [0.214157278609649,0.241289160139954] $
$ 5 $ $9/62$ $ 5/31$ $ [0.586141644350735,0.612800854796395] $
$ 5 $ $21/62$ $11/31$ $[0.387199145203605,0.413858355649265] $
$ 6 $ $1/126$ $1/63$ $[0.786809543609523,0.802555581755556] $
$ 6 $ $61/126$ $31/63$ $[0.197444418244444,0.213190456390477] $
$ 7 $ $1/254$ $1/127$ $[0.803225220690394,0.812352783425512] $
$ 7 $ $125/254$ $63/127$ $[0.187647216574488,0.196774779309606] $
$ 7 $ $17/254$ $9/127$ $[0.699811031164904,0.708527570112261] $
$ 7 $ $109/254$ $55/127$ $[0.291472429887739,0.300188968835096] $
$ 7 $ $41/254$ $21/127$ $[0.576825192903727,0.585555905085145] $
$ 7 $ $85/254$ $43/127$ $[0.414444094914855,0.423174807096273] $
$ 8 $ $1/510$ $1/255$ $[0.812634013261438,0.817780420556863] $
$ 8 $ $253/510$ $127/255$ $[0.182219579443137,0.187365986738562] $
$ 8 $ $73/510$ $37/255$ $[0.613186931037909,0.617835298917647] $
$ 8 $ $181/510$ $91/255$ $[0.382164701082353,0.386813068962091] $
$ 9 $ $1/1022$ $1/511$ $[0.818062650175864,0.820724099383431] $
$ 9 $ $509/1022$ $255/511$ $[0.179275900616569,0.181937349824136] $
$ 9 $ $33/1022$ $17/511$ $[0.755812148539074,0.758473597746640] $
$ 9 $ $477/1022$ $239/511$ $[0.241526402253360,0.244187851460926] $
$ 9 $ $169/1022$ $85/511$ $[0.573835612305023,0.576497061512589] $
$ 9 $ $341/1022$ $171/511$ $[0.423502938487411,0.426164387694977] $
$10 $ $1/2046$ $1/1023$ $[0.821196509738417,0.822528540248941] $
$10 $ $1021/2046$ $511/1023$ $[0.177471459751059,0.178803490261583] $
$10 $ $145/2046$ $73/1023$ $[0.698241698854594,0.699698607225480] $
$10 $ $877/2046$ $439/1023$ $[0.300301392774520,0.301758301145406] $
$11 $ $1/4094$ $1/2047$ $[0.822722890076930,0.823555816343776] $
$11 $ $2045/4094$ $1023/2047$ $[0.176444183656224,0.177277109923070] $
$11 $ $65/4094$ $33/2047$ $[0.786058868717432,0.786683563417567] $
$11 $ $1981/4094$ $991/2047$ $[0.213316436582433,0.213941131282568] $
$11 $ $273/4094$ $137/2047$ $[0.708807402099743,0.709432096799878] $
$11 $ $1773/4094$ $887/2047$ $[0.290567903200122,0.291192597900257] $
$11 $ $585/4094$ $293/2047$ $[0.618230337528651,0.619009737929774] $
$11 $ $1461/4094$ $731/2047$ $[0.380990262070226,0.381769662471349] $
$11 $ $681/4094$ $341/2047$ $[0.572917073341487,0.573541768041622] $
$11 $ $1365/4094$ $683/2047$ $[0.426458231958378,0.427082926658513] $
$12 $ $1/8190$ $1/4095$ $[0.823705054848802,0.824017478548726] $
$12 $ $4093/8190$ $2047/4095$ $[0.175982521451274,0.176294945151198] $
$12 $ $1321/8190$ $661/4095$ $[0.585676663495414,0.585989087195338] $
$12 $ $2773/8190$ $1387/4095$ $[0.414010912804662,0.414323336504586] $
$13 $ $1/16382$ $1/8191$ $[0.824099377662201,0.824366011773877] $
$13 $ $8189/16382$ $4095/8191$ $[0.175633988226123,0.175900622337799] $
$13 $ $129/16382$ $65/8191$ $[0.802834232937408,0.803074203637915] $
$13 $ $8061/16382$ $4031/8191$ $[0.196925796362085,0.197165767062592] $
$13 $ $545/16382$ $273/8191$ $[0.755457525487725,0.755686164238488] $
$13 $ $7645/16382$ $3823/8191$ $[0.244313835761512,0.244542474512275] $
$13 $ $1169/16382$ $585/8191$ $[0.697932644443065,0.698161283193827] $
$13 $ $7021/16382$ $3511/8191$ $[0.301838716806173,0.302067355556935] $
$13 $ $2377/16382$ $1189/8191$ $[0.612842893451498,0.613081331005864] $
$13 $ $5813/16382$ $2907/8191$ $[0.386918668994136,0.387157106548502] $
$13 $ $2729/16382$ $1365/8191$ $[0.572640180643138,0.572864153296945] $
$13 $ $5461/16382$ $2731/8191$ $[0.427135846703055,0.427359819356862] $
Period $s_{\max}-\frac{1}{2}$ $s_{\min}$ $ [c_*,c^*] $
$1$ $-1/2$ $0$ $[0.000000000000000,0.175160000000000]$
$2$ $1/6$ $1/3$ $[0.428133329021334,0.571866670978666]$
$ 3 $ $1/14 $ $ 1/7 $ $ [0.619203577131485,0.697872156658965] $
$ 3 $ $5/14 $ $ 3/7 $ $ [0.302127843341035,0.380796422868515] $
$ 4 $ $1/30 $ $ 1/15 $ $ [0.709633870795466,0.755421357085333] $
$ 4 $ $13/30 $ $ 7/15 $ $ [0.244578642914667,0.290366129204534] $
$ 5 $ $1/62 $ $ 1/31 $ $ [0.758710839860046,0.785842721390351] $
$ 5 $ $29/62$ $15/31 $ $ [0.214157278609649,0.241289160139954] $
$ 5 $ $9/62$ $ 5/31$ $ [0.586141644350735,0.612800854796395] $
$ 5 $ $21/62$ $11/31$ $[0.387199145203605,0.413858355649265] $
$ 6 $ $1/126$ $1/63$ $[0.786809543609523,0.802555581755556] $
$ 6 $ $61/126$ $31/63$ $[0.197444418244444,0.213190456390477] $
$ 7 $ $1/254$ $1/127$ $[0.803225220690394,0.812352783425512] $
$ 7 $ $125/254$ $63/127$ $[0.187647216574488,0.196774779309606] $
$ 7 $ $17/254$ $9/127$ $[0.699811031164904,0.708527570112261] $
$ 7 $ $109/254$ $55/127$ $[0.291472429887739,0.300188968835096] $
$ 7 $ $41/254$ $21/127$ $[0.576825192903727,0.585555905085145] $
$ 7 $ $85/254$ $43/127$ $[0.414444094914855,0.423174807096273] $
$ 8 $ $1/510$ $1/255$ $[0.812634013261438,0.817780420556863] $
$ 8 $ $253/510$ $127/255$ $[0.182219579443137,0.187365986738562] $
$ 8 $ $73/510$ $37/255$ $[0.613186931037909,0.617835298917647] $
$ 8 $ $181/510$ $91/255$ $[0.382164701082353,0.386813068962091] $
$ 9 $ $1/1022$ $1/511$ $[0.818062650175864,0.820724099383431] $
$ 9 $ $509/1022$ $255/511$ $[0.179275900616569,0.181937349824136] $
$ 9 $ $33/1022$ $17/511$ $[0.755812148539074,0.758473597746640] $
$ 9 $ $477/1022$ $239/511$ $[0.241526402253360,0.244187851460926] $
$ 9 $ $169/1022$ $85/511$ $[0.573835612305023,0.576497061512589] $
$ 9 $ $341/1022$ $171/511$ $[0.423502938487411,0.426164387694977] $
$10 $ $1/2046$ $1/1023$ $[0.821196509738417,0.822528540248941] $
$10 $ $1021/2046$ $511/1023$ $[0.177471459751059,0.178803490261583] $
$10 $ $145/2046$ $73/1023$ $[0.698241698854594,0.699698607225480] $
$10 $ $877/2046$ $439/1023$ $[0.300301392774520,0.301758301145406] $
$11 $ $1/4094$ $1/2047$ $[0.822722890076930,0.823555816343776] $
$11 $ $2045/4094$ $1023/2047$ $[0.176444183656224,0.177277109923070] $
$11 $ $65/4094$ $33/2047$ $[0.786058868717432,0.786683563417567] $
$11 $ $1981/4094$ $991/2047$ $[0.213316436582433,0.213941131282568] $
$11 $ $273/4094$ $137/2047$ $[0.708807402099743,0.709432096799878] $
$11 $ $1773/4094$ $887/2047$ $[0.290567903200122,0.291192597900257] $
$11 $ $585/4094$ $293/2047$ $[0.618230337528651,0.619009737929774] $
$11 $ $1461/4094$ $731/2047$ $[0.380990262070226,0.381769662471349] $
$11 $ $681/4094$ $341/2047$ $[0.572917073341487,0.573541768041622] $
$11 $ $1365/4094$ $683/2047$ $[0.426458231958378,0.427082926658513] $
$12 $ $1/8190$ $1/4095$ $[0.823705054848802,0.824017478548726] $
$12 $ $4093/8190$ $2047/4095$ $[0.175982521451274,0.176294945151198] $
$12 $ $1321/8190$ $661/4095$ $[0.585676663495414,0.585989087195338] $
$12 $ $2773/8190$ $1387/4095$ $[0.414010912804662,0.414323336504586] $
$13 $ $1/16382$ $1/8191$ $[0.824099377662201,0.824366011773877] $
$13 $ $8189/16382$ $4095/8191$ $[0.175633988226123,0.175900622337799] $
$13 $ $129/16382$ $65/8191$ $[0.802834232937408,0.803074203637915] $
$13 $ $8061/16382$ $4031/8191$ $[0.196925796362085,0.197165767062592] $
$13 $ $545/16382$ $273/8191$ $[0.755457525487725,0.755686164238488] $
$13 $ $7645/16382$ $3823/8191$ $[0.244313835761512,0.244542474512275] $
$13 $ $1169/16382$ $585/8191$ $[0.697932644443065,0.698161283193827] $
$13 $ $7021/16382$ $3511/8191$ $[0.301838716806173,0.302067355556935] $
$13 $ $2377/16382$ $1189/8191$ $[0.612842893451498,0.613081331005864] $
$13 $ $5813/16382$ $2907/8191$ $[0.386918668994136,0.387157106548502] $
$13 $ $2729/16382$ $1365/8191$ $[0.572640180643138,0.572864153296945] $
$13 $ $5461/16382$ $2731/8191$ $[0.427135846703055,0.427359819356862] $
[1]

Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229

[2]

Joshua P. Bowman, Slade Sanderson. Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces. Journal of Modern Dynamics, 2020, 16: 109-153. doi: 10.3934/jmd.2020005

[3]

Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483

[4]

M. Baake, P. Gohlke, M. Kesseböhmer, T. Schindler. Scaling properties of the Thue–Morse measure. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4157-4185. doi: 10.3934/dcds.2019168

[5]

David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287

[6]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684

[7]

Jeanette Olli. Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4173-4186. doi: 10.3934/dcds.2013.33.4173

[8]

Oliver Jenkinson. Ergodic Optimization. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197

[9]

Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178

[10]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383

[11]

Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829

[12]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[13]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501

[14]

Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, 2021, 29 (4) : 2819-2827. doi: 10.3934/era.2021015

[15]

Oliver Jenkinson. Optimization and majorization of invariant measures. Electronic Research Announcements, 2007, 13: 1-12.

[16]

Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149

[17]

Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048

[18]

Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249

[19]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190

[20]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (234)
  • HTML views (111)
  • Cited by (0)

Other articles
by authors

[Back to Top]