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 and 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.

[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), 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.(3), 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.

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.

[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), 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.(3), 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.

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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