The complexity of permutive cellular automata Jung-Chao Ban a,b , Chih-Hung Chang c , Ting-Ju Chen a , Mei-Shao Lin a a Department of Applied Mathematics, National Dong Hwa University, Hualian 97063, Taiwan, R.O.C. b Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan, R.O.C. c Department of Applied Mathematics, National Pingtung University of Education, Pingtung 90003, Taiwan, R.O.C. Abstract This paper studies cellular automata in two aspects: Ergodic and topological behavior. For ergodic aspect, the formulae of measure-theoretic entropy, topological entropy and topological pressure are given in closed forms and Parry measure is demonstrated to be an equilibrium measure for some potential function. For topological aspect, an example is examined to show that the exhibition of snap-back repellers for a cellular automaton infers Li-Yorke chaos. In addition, bipermutive cellular automata are optimized for the exhibition of snap-back repellers in permutive cellular automata whenever two-sided shift space is considered. Key words: Cellular automata, permutive, equilibrium measure, Parry measure, snap-back repeller, chaos 2000 MSC: 28D20, 37B15, 37B40, 47A35 1. Introduction Cellular automaton (CA), introduced by Ulam [1] and von Neumann [2] as a model for self-production, is a particular class of dynamical systems which is defined by a local rule acting on a discrete space and is widely studied in a variety of contexts in physics, biology and computer science. CA is brought to the attention of wide audience through an ecological model, say Conway’s Game-of-Life, which consists of simple rule but leads to complex behavior. Physicists use CA to investigate many phenomena such as spiral waves, phase transi- tions, reaction-diffusion processes, fluid, and so on. In 1980s, Hardy, Pazzis and Pomeau introduced the so-called HPP lattice gas model to study fundamental statistical prop- erties of a gas of interacting particles. Greenberg and Hastings yield CA to investi- gate reaction-diffusion equations and find that such a simplified model can still gen- erate spatial-temporal structures similar to the original system. Reader is referred to [4, 5, 6, 7, 8, 9, 10, 3, 11, 12, 13, 14, 15] and references therein for more details. Email addresses: jcban@mail.ndhu.edu.tw (Jung-Chao Ban), chchang@mail.npue.edu.tw (Chih-Hung Chang), 94a1028@stu.nhlue.edu.tw (Ting-Ju Chen), 94a1029@stu.nhlue.edu.tw (Mei-Shao Lin)