Ergodic properties of certain surjective cellular automata

We consider one-dimensional cellular automata, i.e. the mapsT:Pℤ→Pℤ (P is a finite set with more than one element) which are given by (Tx)i=F(xi+l, ...,xi+r),x=(xi)∈ℤ∈Pℤ for some integersl≤r and a mappingF∶Pr−l+1→P. We prove that ifF is right- (left-) permutative (in Hedlund's terminology) and 0≤l<r (resp.l<r≤0), then the natural extension of the dynamical system (Pℤ, ℬ, μ,T) is a Bernoulli automorphism (μ stands for the (1/p, ..., 1/p)-Bernoulli measure on the full shiftPℤ). Ifr<0 orl>0 andT is surjective, then the natural extension of the system (Pℤ, ℬ, μ,T) is aK-automorphism. We also prove that the shift ℤ2-action on a two-dimensional subshift of finite type canonically associated with the cellular automatonT is mixing, ifF is both right and left permutative. These results answer some questions raised in [SR].