# American Institute of Mathematical Sciences

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

## 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

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.

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

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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.  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
 $(a_1, a_2, a_3)$ 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 $a_2=1$ $D_2=0$
 $(a_1, a_2, a_3)$ 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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