$ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization

Aihua Fan; Jörg Schmeling; Weixiao Shen

doi:10.3934/dcds.2020363

Article Contents

2021, Volume 41, Issue 1: 297-327. Doi: 10.3934/dcds.2020363

This issue Previous Article Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions Next Article On $ \epsilon $-escaping trajectories in homogeneous spaces

$ 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

Early access: October 30, 2020

Published online: October 30, 2020

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary:11L07, 37A05, 37B10.

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Figure 1. The graphs of $ f_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $

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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)

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Figure 3. The graphs of $ \log|2\sin \pi (x-b)| $ on the intervals $ [0,1] $ and $ [b,b+1] $ with $ b = 1/3 $

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Figure 4. The graphs of $ f'_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $

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Figure 5. The branch $ T|_{C_\lambda} $

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Figure 6. The graphs of $ e_0 $ and $ e_{1/4} $

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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.

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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] $

| Show Table

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