Characterization of noncorrelated pattern sequences and correlation dimensions

Yu Zheng; Li Peng; Teturo Kamae

doi:10.3934/dcds.2018223

Article Contents

2018, Volume 38, Issue 10: 5085-5103. Doi: 10.3934/dcds.2018223

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Characterization of noncorrelated pattern sequences and correlation dimensions

1.
Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China
2.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
3.
Advanced Mathematical Institute, Osaka City University, Osaka, 558-8585, Japan

^* Corresponding author: Li Peng
^* Corresponding author: Li Peng

Received: December 2017

Revised: May 2018

Published: October 2018

This work was supported by the NSFC [11571127, 11431007].

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Abstract

We consider the correlation functions of binary pattern sequences of degree 3 as well as those with general degrees and special patterns and obtain necessary and sufficient conditions to be noncorrelated. We also obtain the correlation dimensions for those with degree 2.

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Mathematics Subject Classification: Primary: 47B15; Secondary: 11B50.

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Table 1.

	$(a_1, a_2, a_3)$	pattern set	spectrum
	$(-1, 1, 1)$	$\{01, 10, 11\}$
$a_1a_2a_3=-1$	$(1, -1, 1)$	$\{01\}$	noncorrelated sequence
	$(1, 1, -1)$	$\{11\}$	$D_2=1$
	$(-1, -1, -1)$	$\{10\}$
$a_1a_2a_3=1$	$(1, -1, -1)$	$\{01, 11\}$	singular spectrum
$a_2=-1$	$(-1, -1, 1)$	$\{10, 11\}$	$D_2=3-\log_2(1+\sqrt{17})$
$a_1a_2a_3=1$	$(-1, 1, -1)$	$\{01, 10\}$	periodic sequence
$a_2=1$			$D_2=0$

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References

[1]	J.-P. Allouche and P. Liardet, Generalized Rudin-Shapiro sequences, Acta Arith., 60 (1991), 1-27. doi: 10.4064/aa-60-1-1-27.
[2]	D. W. Boyd, J. H. Cook and P. Morton, On sequences of $±$1's defined by binary patterns, Dissertationes Math., 283 (1989), 64pp.
[3]	J. Coquet, T. Kamae and M. Mendès France, Sur la mesure spectrale de certaines suites arithmétiques, Bull. Soc. Math. France, 105 (1977), 369-384.
[4]	N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002.
[5]	C. Godrèche and J. M. Luck, Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures, J. Phys. A, 23 (1990), 3769-3797. doi: 10.1088/0305-4470/23/16/024.
[6]	P. Morton, Connections between binary patterns and paperfolding, Sém. Théor. Nombres Bordeaux, 2 (1990), 1-12. doi: 10.5802/jtnb.16.
[7]	P. Morton and W. J. Mourant, Paper folding, digit patterns and groups of arithmetic fractals, Proc. London Math. Soc., 59 (1989), 253-293. doi: 10.1112/plms/s3-59.2.253.
[8]	M. Niu and Z. X. Wen, Correlation dimension of the spectral measure for m-multiplicative sequences, (Chinese), Acta Math. Sci. Ser. A (Chin. Ed.), 27 (2007), 862-870.
[9]	L. Peng and T. Kamae, Spectral measure of the Thue-Morse sequence and the dynamical system and random walk related to it, Ergodic Theory Dynam. Systems, 36 (2016), 1247-1259. doi: 10.1017/etds.2014.121.
[10]	K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, 1983.
[11]	M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Springer-Verlag, Berlin, 1987.
[12]	M. A. Zaks, A. S. Pikovsky and J. Kurths, On the correlation dimension of the spectral measure for the Thue-Morse sequence, J. Statist. Phys., 88 (1997), 1387-1392. doi: 10.1007/BF02732440.