Some Properties of topological pressure on cellular automata

Year 2014, Volume: 1 Issue: 1, 41 - 51, 01.03.2014

Chih-Hung Chang

https://doi.org/10.13069/jacodesmath.66382

Cited By: 1

Abstract

This paper investigates the ergodicity and the power rule of the topological pressure of a cellular automaton. If a cellular automaton is either leftmost or rightmost premutive (due to the terminology given by Hedlund [Math.~Syst.~Theor.~3, 320-375, 1969]), then it is ergodic with respect to the uniform Bernoulli measure. More than that, the relation of topological pressure between the original cellular automaton and its power rule is expressed in a closed form. As an application, the topological pressure of a linear cellular automaton can be computed explicitly.

Keywords

Cellular automata, Permutive, Ergodicity, Topological pressure

References

• H. Akin, On the ergodic properties of certain additive cellular automata over Zm, Appl. Math. Comput., 168, 192-197, 2005.
• H. Akin, Some strong ergodic properties of 1d invertible linear cellular automata, arXiv: 0902.3762, 20
• H. Akin, J.-C. Ban, and C.-H. Chang, On the qualitative behavior of linear cellular automata, J. Cell. Autom., 8, 205-231, 2013.
• J.-C. Ban and C.-H. Chang, The topological pressure of linear cellular automata, Entropy, 11, 271- 284, 2009.
• J.-C. Ban, C.-H. Chang, T.-J. Chen, and M.-S. Lin, The complexity of permutive cellular automata, J. Cell. Autom., 6, 385-397, 2011.
• G. Bub, A. Shrier, and L. Glass, Spiral wave generation in heterogeneous excitable media, Phys. Rev. Lett., 88(5), 058101, 2002.
• Y. B. Chernyak, A. B. Feldman, and R. J. Cohen, Correspondence between discrete and continuous models of excitable media: Trigger waves, Phys. Rev. E, 55, 3125-3233, 1997.
• E. M. Coven, Topological entropy of block maps, Proc. Amer. Math. Soc., 78, 590-594, 1980.
• E. M. Coven and M. Paul, Endomorphisms of irreducible shifts of finite type, Math. Syst. Theory, 8, 165-177, 1974.
• A. B. Feldman, Y. B. Chernyak, and R. J. Cohen, Wave-front propagation in a discrete model of excitable media, Phys. Rev. E, 57, 7025-7040, 1998.
• J. M. Greenberg, C. Greene, and S. P. Hastings, Acombinatorial problem arising in the study of reaction-diffusion equations, SIAM J. Algebraic Discrete Methods, 1, 34-42, 1980.
• J. M. Greenberg, B. D. Hassard, and S. P. Hastings, Pattern formation and periodic structure in systems modeled by reaction-diffusion equations, Bull. Amer. Math. Soc., 84, 1296-1327, 1978.
• J. M. Greenberg and S. P. Hastings, Spatial patterns for discrete models of diffusion in excitable media, SIAM J. Appl. Math., 34, 515-523, 1978.
• J. Hardy, O. de Pazzis, and Y. Pomeau, Molecular dynamics of a classical lattice gas: transport properties and time correlation functions, Phys. Rev. A, 13, 1949-1960, 1976.
• G. A. Hedlund, Endomorphisms and automorphisms of full shift dynamical system, Math. Systems Theory, 3, 320-375, 1969.
• D. Richardson, Tessellation with local transformations, J. Compuut. System Sci., 6, 373-388, 1972.
• M. Sabbagan and P. Nasertayoob, Ergodic theory and dynamical systems from a physical point of view, Arab. J. Sci. Eng. Sect. A Sci., 33, 373-387, 2008.
• M. A. Shereshevsky, Ergodic properties of certain surjective cellular automata, Monatsh. Math., 114, 305-316, 1992.
• M. Shirvani and T. D. Rogers, On ergodic one-dimensional cellular automata, Commun. Math. Phys., 136, 599-605, 1991.
• A. R. Smith III, Simple computational universal spaces, J. Assoc. Comput. Mach., 18, 339-353, 1971.
• S. Ulam, Random process and transformations, Proc. Int. Congress of Math., 2, 264-275, 1952.
• G. Y. Vichniac, Boolean derivatives on cellular automata, Phys. D, 45, 63-74, 1990.
• J. von Neumann, Theory of self-reproducing automata, Univ. of Illinois Press, Urbana, 1966.
• P. Walters, An introduction to ergodic theory, Springer-Verlag New York, 1982.
• T. Ward, Additive cellular automata and volume growth, Entropy, 2, 142-167, 2000.
• J. R. Weimar, J. J. Tyson, and L. T. Watson, Third generation cellular automaton for modeling excitable media, Phys. D, 55, 328-339, 1992.
• S. Wolfram, Statistical mechanics of cellular automata, Rev. Modern Physics, 55, 601-644, 1983.