# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705
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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. Google Scholar [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). doi: 10.1090/S0065-9266-2012-00653-8. Google Scholar [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,, , (). Google Scholar [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. doi: 10.3934/dcds.2005.13.637. Google Scholar [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. doi: 10.4310/AJM.2007.v11.n3.a7. Google Scholar [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. doi: 10.1007/s00605-002-0486-z. Google Scholar [7] M. Boyle, R. Pavlov and M. Schraudner, Multidimensional sofic shifts without separation and their factors,, Trans. Amer. Math. Soc., 362 (2010), 4617. doi: 10.1090/S0002-9947-10-05003-8. Google Scholar [8] G. D. Chakerian and S. K. Stein, Some intersection properties of convex bodies,, Proc. Amer. Math. Soc., 18 (1967), 109. doi: 10.1090/S0002-9939-1967-0206818-3. Google Scholar [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. Google Scholar [10] M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type,, Annals of Mathematics, 171 (2010), 2011. doi: 10.4007/annals.2010.171.2011. Google Scholar [11] W. G. Hu and S. S. Lin, Nonemptiness problems of plane square tiling with two colors,, Proc. Amer. Math. Soc., 139 (2011), 1045. doi: 10.1090/S0002-9939-2010-10518-X. Google Scholar [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. doi: 10.1016/j.jfa.2011.04.014. Google Scholar [13] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar [14] E. Lindenstrauss and B. Weiss, Mean topological dimension,, Israel J. Math., 115 (2000), 1. doi: 10.1007/BF02810577. Google Scholar [15] N. G. Markley and M. E. Paul, Maximal measures and entropy for $Z^{\nu}$ subshift of finite type,, Classical Mechanics and Dynamical Systems, 70 (1981), 135. Google Scholar [16] N. G. Markley and M. E. Paul, Matrix subshifts for $Z^{\nu }$ symbolic dynamics,, Proc. London Math. Soc., 43 (1981), 251. doi: 10.1112/plms/s3-43.2.251. Google Scholar [17] P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982). doi: 10.1007/978-1-4612-5775-2. Google Scholar
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