### Dynamics groups of asynchronous cellular automata

*Matthew Macauley*

^{}

*, Jon McCammond*

^{}

*and Henning S. Mortveit*

^{}

^{}

**DOI: **10.1007/s10801-010-0231-y

#### Abstract

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is

*π*-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the*dynamics group*. We have previously shown that exactly 104 of the possible 2^{2 3}=256 2^{2^{3}}=256 cellular automaton rules are*π*-independent. In the article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.**Pages: **11–35

**Keywords: **keywords sequential dynamical systems; cellular automata; update order; dynamics groups; Coxeter groups; periodic points; Fibonacci numbers; Lucas numbers

Full Text: PDF

#### References

1. Hansson, A.Å., Mortveit, H.S., Reidys, C.M.: On asynchronous cellular automata. Adv. Complex Syst. 8, 521-538 (2005)

2. Macauley, M., McCammond, J., Mortveit, H.S.: Order independence in asynchronous cellular automata. J. Cell. Autom. 3, 37-56 (2008)

3. Martin, O., Odlyzko, A.M., Wolfram, S.: Algebraic properties of cellular automata. Commun. Math. Phys. 93, 219-258 (1984)

4. Mortveit, H.S., Reidys, C.M.: An Introduction to Sequential Dynamical Systems. Springer, Berlin (2007)

5. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Published electronically at:

6. Stanley, R.P.: Enumerative Combinatorics, vol.

1. Cambridge University Press, Cambridge (1997)

7. Weir, A.J.: Sylow p-subgroups of the general linear group over finite fields of characteristic p. Proc.

2. Macauley, M., McCammond, J., Mortveit, H.S.: Order independence in asynchronous cellular automata. J. Cell. Autom. 3, 37-56 (2008)

3. Martin, O., Odlyzko, A.M., Wolfram, S.: Algebraic properties of cellular automata. Commun. Math. Phys. 93, 219-258 (1984)

4. Mortveit, H.S., Reidys, C.M.: An Introduction to Sequential Dynamical Systems. Springer, Berlin (2007)

5. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Published electronically at:

6. Stanley, R.P.: Enumerative Combinatorics, vol.

1. Cambridge University Press, Cambridge (1997)

7. Weir, A.J.: Sylow p-subgroups of the general linear group over finite fields of characteristic p. Proc.

© 1992–2009 Journal of Algebraic Combinatorics

©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition