American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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), vi+60 pp. 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), 637-658. 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), 497-534. 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), 95-110. 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), 4617-4653. 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), 109-112. 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), 109-178.  Google Scholar

[10]

M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Annals of Mathematics, 171 (2010), 2011-2038. 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), 1045-1059. 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), 1028-1082. 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, Cambridge, 1995. doi: 10.1017/CBO9780511626302.  Google Scholar

[14]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. 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, Lecture Notes in Pure and Appl. Math., 70 (1981), 135-157.  Google Scholar

[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.  Google Scholar

[17]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5775-2.  Google Scholar

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.  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), vi+60 pp. 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), 637-658. 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), 497-534. 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), 95-110. 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), 4617-4653. 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), 109-112. 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), 109-178.  Google Scholar

[10]

M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Annals of Mathematics, 171 (2010), 2011-2038. 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), 1045-1059. 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), 1028-1082. 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, Cambridge, 1995. doi: 10.1017/CBO9780511626302.  Google Scholar

[14]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. 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, Lecture Notes in Pure and Appl. Math., 70 (1981), 135-157.  Google Scholar

[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.  Google Scholar

[17]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5775-2.  Google Scholar

[1]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191
[2]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[3]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[4]

Michael Schraudner. Projectional entropy and the electrical wire shift. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 333-346. doi: 10.3934/dcds.2010.26.333
[5]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
[6]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[7]

Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507
[8]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[9]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[10]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[11]

Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
[12]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[13]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[14]

Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021131
[15]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[16]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[17]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[18]

Jérôme Rousseau, Paulo Varandas, Yun Zhao. Entropy formulas for dynamical systems with mistakes. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4391-4407. doi: 10.3934/dcds.2012.32.4391
[19]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[20]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy