TY - JOUR

T1 - Finite entropy for multidimensional cellular automata

AU - Meyerovitch, Tom

N1 - Funding Information:
During the past three years, 45 engineers, scientists and support personnel have worked hard at JPL to bring the AVIRIS system to completion. The author would like to express his deep gratitude to all those whose efforts have contributed to the success of this joint endeavor. The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

PY - 2008/8/1

Y1 - 2008/8/1

N2 - Let X = S where is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T:X → X (continuous, commuting with the action of ). Shereshevsky [Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag. Math. (N.S.) 4(2) (1993), 203-210] proved that for =ℤd with d > 1 no CA can be forward expansive, raising the following conjecture: for G = ℤd, d > 1, the topological entropy of any CA is either zero or infinite. Morris and Ward [Entropy bounds for endomorphisms commuting with K actions. Israel J. Math. 106 (1998), 1-11] proved this for linear CAs, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exists a d-dimensional CA with finite, non-zero topological entropy. We also discuss a measure-theoretic counterpart of this question for measure-preserving CAs.

AB - Let X = S where is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T:X → X (continuous, commuting with the action of ). Shereshevsky [Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag. Math. (N.S.) 4(2) (1993), 203-210] proved that for =ℤd with d > 1 no CA can be forward expansive, raising the following conjecture: for G = ℤd, d > 1, the topological entropy of any CA is either zero or infinite. Morris and Ward [Entropy bounds for endomorphisms commuting with K actions. Israel J. Math. 106 (1998), 1-11] proved this for linear CAs, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exists a d-dimensional CA with finite, non-zero topological entropy. We also discuss a measure-theoretic counterpart of this question for measure-preserving CAs.

UR - http://www.scopus.com/inward/record.url?scp=47249154761&partnerID=8YFLogxK

U2 - 10.1017/S0143385707000855

DO - 10.1017/S0143385707000855

M3 - Article

AN - SCOPUS:47249154761

VL - 28

SP - 1243

EP - 1260

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -