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: November 30, 2017

Revised: April 30, 2018

Published: September 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

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