Ergodic properties of certain surjective cellular automata

We consider one-dimensional cellular automata, i.e. the mapsT:P ℤ →P ℤ (P is a finite set with more than one element) which are given by (Tx) i =F(x i+l , ...,x i+r ),x=(x i )∈ℤ∈P ℤ for some integersl≤r and a mappingF∶P r−l+1→P. We prove that ifF is right- (left-) permutative (in Hedlund's terminology) and 0≤l0 andT is surjective, then the natural extension of the system (P ℤ , 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].