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

On spatial entropy of multi-dimensional symbolic dynamical systems

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

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.
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, arXiv:1112.2471.

[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.

show all references

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, arXiv:1112.2471.

[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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