American Institute of Mathematical Sciences

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October  2018, 38(10): 5085-5103. doi: 10.3934/dcds.2018223

Characterization of noncorrelated pattern sequences and correlation dimensions

Yu Zheng 1,2, , Li Peng 1,, and  Teturo Kamae 3,

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

Received  December 2017 Revised  May 2018 Published  July 2018

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

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.

Keywords: Fractal geometry, dynamical system, pattern sequence, correlation measure, correlation dimension.
Mathematics Subject Classification: Primary: 47B15; Secondary: 11B50.
Citation: Yu Zheng, Li Peng, Teturo Kamae. Characterization of noncorrelated pattern sequences and correlation dimensions. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5085-5103. doi: 10.3934/dcds.2018223
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.  Google Scholar

[2]

D. W. Boyd, J. H. Cook and P. Morton, On sequences of $±$1's defined by binary patterns, Dissertationes Math., 283 (1989), 64pp.  Google Scholar

[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.   Google Scholar

[4]

N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002. Google Scholar

[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.  Google Scholar

[6]

P. Morton, Connections between binary patterns and paperfolding, Sém. Théor. Nombres Bordeaux(2), 2 (1990), 1-12.  doi: 10.5802/jtnb.16.  Google Scholar

[7]

P. Morton and W. J. Mourant, Paper folding, digit patterns and groups of arithmetic fractals, Proc. London Math. Soc.(3), 59 (1989), 253-293.  doi: 10.1112/plms/s3-59.2.253.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, 1983. Google Scholar

[11]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Springer-Verlag, Berlin, 1987. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

D. W. Boyd, J. H. Cook and P. Morton, On sequences of $±$1's defined by binary patterns, Dissertationes Math., 283 (1989), 64pp.  Google Scholar

[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.   Google Scholar

[4]

N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002. Google Scholar

[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.  Google Scholar

[6]

P. Morton, Connections between binary patterns and paperfolding, Sém. Théor. Nombres Bordeaux(2), 2 (1990), 1-12.  doi: 10.5802/jtnb.16.  Google Scholar

[7]

P. Morton and W. J. Mourant, Paper folding, digit patterns and groups of arithmetic fractals, Proc. London Math. Soc.(3), 59 (1989), 253-293.  doi: 10.1112/plms/s3-59.2.253.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, 1983. Google Scholar

[11]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Springer-Verlag, Berlin, 1987. Google Scholar

[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.  Google Scholar

Table 1.   
$(a_1, a_2, a_3)$pattern setspectrum
$(-1, 1, 1)$ $\{01, 10, 11\}$
$a_1a_2a_3=-1$ $(1, -1, 1)$ $\{01\}$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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$(a_1, a_2, a_3)$pattern setspectrum
$(-1, 1, 1)$ $\{01, 10, 11\}$
$a_1a_2a_3=-1$ $(1, -1, 1)$ $\{01\}$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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