On spatial entropy of multi-dimensional symbolic dynamical systems

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July 2016, 36(7): 3705-3717. doi: 10.3934/dcds.2016.36.3705

On spatial entropy of multi-dimensional symbolic dynamical systems

Wen-Guei Hu ^1, and Song-Sun Lin ^2,

1.	College of Mathematics, Sichuan University, Chengdu 610064, China
2.	Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300

Received March 2015 Revised December 2015 Published March 2016

Abstract
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The commonly used topological entropy $h_{top}(\mathcal{U})$ of the multi-dimensional shift space $\mathcal{U}$ is the rectangular spatial entropy $h_{r}(\mathcal{U})$ which is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space $\mathbb{Z}^{d}$, $d\geq 2$. This work studies spatial entropy $h_{\Omega}(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding system $\Omega=\{\Omega(n)\}_{n=1}^{\infty}$ where $\Omega(n)$ is increasing finite sublattices and expands to $\mathbb{Z}^{d}$. $\Omega$ is called genuinely $d$-dimensional if $\Omega(n)$ contains no lower-dimensional part whose size is comparable to that of its $d$-dimensional part. We show that $h_{r}(\mathcal{U})$ is the supremum of $h_{\Omega}(\mathcal{U})$ for all genuinely $d$-dimensional $\Omega$. Furthermore, when $\Omega$ is genuinely $d$-dimensional and satisfies certain conditions, then $h_{\Omega}(\mathcal{U})=h_{r}(\mathcal{U})$. On the contrary, when $\Omega(n)$ contains a lower-dimensional part which is comparable to its $d$-dimensional part, then $h_{r}(\mathcal{U}) < h_{\Omega}(\mathcal{U})$ for some $\mathcal{U}$. Therefore, $h_{r}(\mathcal{U})$ is appropriate to be the $d$-dimensional spatial entropy.

Keywords: spatial entropy, symbolic dynamical system, shift space, Topological entropy, block gluing..

Mathematics Subject Classification: Primary: 37B40, 37B10, 28D20; Secondary: 37B5.

Citation: Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705

References:

[1]	P. Ballister, B. Bollobás and A. Quas, Entropy Along Convex Shapes, Random Tilings and Shifts of Finite Type, Illinois journal of Matlaematics, 46 (2002), 781-795.
[2]	J. C. Ban, W. G. Hu, S. S. Lin and Y. H. Lin, Zeta functions for two-dimensional shifts of finite type, Memo. Amer. Math. Soc., 221 (2013), vi+60 pp. doi: 10.1090/S0065-9266-2012-00653-8.
[3]	J. C. Ban, W. G. Hu, S. S. Lin and Y. H. Lin, Verification of mixing properties in two-dimensional shifts of finite type, submitted,, , ().
[4]	J. C. Ban and S. S. Lin, Patterns generation and transition matrices in multi-dimensional lattice models, Discrete Contin. Dyn. Syst., 13 (2005), 637-658. doi: 10.3934/dcds.2005.13.637.
[5]	J. C. Ban, S. S. Lin and Y. H. Lin, Patterns generation and spatial entropy in two dimensional lattice models, Asian J. Math., 11 (2007), 497-534. doi: 10.4310/AJM.2007.v11.n3.a7.
[6]	K. Böröczky Jr., M. A. Hernández Cifre and G. Salinas, Optimizing area and perimeter of convex sets for fixed circumradius and inradius, Monatsh. Math., 138 (2003), 95-110. doi: 10.1007/s00605-002-0486-z.
[7]	M. Boyle, R. Pavlov and M. Schraudner, Multidimensional sofic shifts without separation and their factors, Trans. Amer. Math. Soc., 362 (2010), 4617-4653. doi: 10.1090/S0002-9947-10-05003-8.
[8]	G. D. Chakerian and S. K. Stein, Some intersection properties of convex bodies, Proc. Amer. Math. Soc., 18 (1967), 109-112. doi: 10.1090/S0002-9939-1967-0206818-3.
[9]	S. N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4 (1996), 109-178.
[10]	M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Annals of Mathematics, 171 (2010), 2011-2038. doi: 10.4007/annals.2010.171.2011.
[11]	W. G. Hu and S. S. Lin, Nonemptiness problems of plane square tiling with two colors, Proc. Amer. Math. Soc., 139 (2011), 1045-1059. doi: 10.1090/S0002-9939-2010-10518-X.
[12]	W. Huang, X. D. Ye and G. H. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082. doi: 10.1016/j.jfa.2011.04.014.
[13]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[14]	E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. doi: 10.1007/BF02810577.
[15]	N. G. Markley and M. E. Paul, Maximal measures and entropy for $Z^{\nu}$ subshift of finite type, Classical Mechanics and Dynamical Systems, Lecture Notes in Pure and Appl. Math., 70 (1981), 135-157.
[16]	N. G. Markley and M. E. Paul, Matrix subshifts for $Z^{\nu }$ symbolic dynamics, Proc. London Math. Soc., 43 (1981), 251-272. doi: 10.1112/plms/s3-43.2.251.
[17]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5775-2.

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References:

[1]	P. Ballister, B. Bollobás and A. Quas, Entropy Along Convex Shapes, Random Tilings and Shifts of Finite Type, Illinois journal of Matlaematics, 46 (2002), 781-795.
[2]	J. C. Ban, W. G. Hu, S. S. Lin and Y. H. Lin, Zeta functions for two-dimensional shifts of finite type, Memo. Amer. Math. Soc., 221 (2013), vi+60 pp. doi: 10.1090/S0065-9266-2012-00653-8.
[3]	J. C. Ban, W. G. Hu, S. S. Lin and Y. H. Lin, Verification of mixing properties in two-dimensional shifts of finite type, submitted,, , ().
[4]	J. C. Ban and S. S. Lin, Patterns generation and transition matrices in multi-dimensional lattice models, Discrete Contin. Dyn. Syst., 13 (2005), 637-658. doi: 10.3934/dcds.2005.13.637.
[5]	J. C. Ban, S. S. Lin and Y. H. Lin, Patterns generation and spatial entropy in two dimensional lattice models, Asian J. Math., 11 (2007), 497-534. doi: 10.4310/AJM.2007.v11.n3.a7.
[6]	K. Böröczky Jr., M. A. Hernández Cifre and G. Salinas, Optimizing area and perimeter of convex sets for fixed circumradius and inradius, Monatsh. Math., 138 (2003), 95-110. doi: 10.1007/s00605-002-0486-z.
[7]	M. Boyle, R. Pavlov and M. Schraudner, Multidimensional sofic shifts without separation and their factors, Trans. Amer. Math. Soc., 362 (2010), 4617-4653. doi: 10.1090/S0002-9947-10-05003-8.
[8]	G. D. Chakerian and S. K. Stein, Some intersection properties of convex bodies, Proc. Amer. Math. Soc., 18 (1967), 109-112. doi: 10.1090/S0002-9939-1967-0206818-3.
[9]	S. N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4 (1996), 109-178.
[10]	M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Annals of Mathematics, 171 (2010), 2011-2038. doi: 10.4007/annals.2010.171.2011.
[11]	W. G. Hu and S. S. Lin, Nonemptiness problems of plane square tiling with two colors, Proc. Amer. Math. Soc., 139 (2011), 1045-1059. doi: 10.1090/S0002-9939-2010-10518-X.
[12]	W. Huang, X. D. Ye and G. H. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082. doi: 10.1016/j.jfa.2011.04.014.
[13]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[14]	E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. doi: 10.1007/BF02810577.
[15]	N. G. Markley and M. E. Paul, Maximal measures and entropy for $Z^{\nu}$ subshift of finite type, Classical Mechanics and Dynamical Systems, Lecture Notes in Pure and Appl. Math., 70 (1981), 135-157.
[16]	N. G. Markley and M. E. Paul, Matrix subshifts for $Z^{\nu }$ symbolic dynamics, Proc. London Math. Soc., 43 (1981), 251-272. doi: 10.1112/plms/s3-43.2.251.
[17]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5775-2.

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